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 _Z(t**Zh6y
function z = zernfun(n,m,r,theta,nflag) @ZU$W9g
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. JwbZ`Z*w
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j Mn,N9Mf
% and angular frequency M, evaluated at positions (R,THETA) on the SAdT#0J
% unit circle. N is a vector of positive integers (including 0), and k&ooV4#f6
% M is a vector with the same number of elements as N. Each element @$jV"Y
% k of M must be a positive integer, with possible values M(k) = -N(k) 9nN1f@Y
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, lQ?jdi
% and THETA is a vector of angles. R and THETA must have the same ~)f^y!PMQ
% length. The output Z is a matrix with one column for every (N,M) bg Ux&3
% pair, and one row for every (R,THETA) pair. zx%WV@O9
% Rq}lW.<r
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Mw0>p5+ cy
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *,JE[M
% with delta(m,0) the Kronecker delta, is chosen so that the integral :e7\z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @[lMh9`
% and theta=0 to theta=2*pi) is unity. For the non-normalized AtJ{d^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o=R(DK# U
% [eD0L71[
% The Zernike functions are an orthogonal basis on the unit circle. fz^j3'!\
% They are used in disciplines such as astronomy, optics, and 5;}W=x^$a
% optometry to describe functions on a circular domain. 0S4BV%7F
% ^($'l)I
% The following table lists the first 15 Zernike functions. ~uc7R/3ss
% d 5yEgc;z
% n m Zernike function Normalization D#LV&4e>.E
% -------------------------------------------------- l$/pp
% 0 0 1 1 2yK">xYY@
% 1 1 r * cos(theta) 2 Y]SF0:v!n
% 1 -1 r * sin(theta) 2
;v.[aq
% 2 -2 r^2 * cos(2*theta) sqrt(6) VVDN3
% 2 0 (2*r^2 - 1) sqrt(3) j"nOxs
% 2 2 r^2 * sin(2*theta) sqrt(6) 83c2y;|8
% 3 -3 r^3 * cos(3*theta) sqrt(8) nmU1xv_
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) lBbb7*Ljt<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) }R}+8
% 3 3 r^3 * sin(3*theta) sqrt(8) (1'DZxJ&u
% 4 -4 r^4 * cos(4*theta) sqrt(10) LJ[zF~4#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _P.I+!w:x
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) o^/
#i`)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F_*']:p
% 4 4 r^4 * sin(4*theta) sqrt(10) [@Ac#
% -------------------------------------------------- nW)+-Wxq
% uHI(-!O
% Example 1: w1G(s$;C
% 2Xl+}M.:Y
% % Display the Zernike function Z(n=5,m=1) $Er=i }`
% x = -1:0.01:1; =#u4^%i)
% [X,Y] = meshgrid(x,x); !ekByD
% [theta,r] = cart2pol(X,Y); [8Pt$5]^
% idx = r<=1; *Y(59J2
% z = nan(size(X)); Ow4 _0l&
% z(idx) = zernfun(5,1,r(idx),theta(idx)); FC1rwXL(
% figure ]u5TvI,C
% pcolor(x,x,z), shading interp Em(_W5
ND{
% axis square, colorbar <gwRE{6U
% title('Zernike function Z_5^1(r,\theta)') !Axe}RD'
% W p)!G
% Example 2: ipn-HUrE@
% `9r{z;UQ
% % Display the first 10 Zernike functions U"7o;q
% x = -1:0.01:1; |Ml~Pmpp
% [X,Y] = meshgrid(x,x); 9F807G\4Qt
% [theta,r] = cart2pol(X,Y); dq '2y
% idx = r<=1; .h W>#
% z = nan(size(X)); %k#+nad
% n = [0 1 1 2 2 2 3 3 3 3]; q8$t4_pF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; P7-k!p"
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Ve(<s
% y = zernfun(n,m,r(idx),theta(idx)); iW6O9~
% figure('Units','normalized') \na$Sb+
% for k = 1:10 ;$iT]S
% z(idx) = y(:,k); sg,\!'
% subplot(4,7,Nplot(k)) Ln#o:" E
% pcolor(x,x,z), shading interp 5}G_2<G
% set(gca,'XTick',[],'YTick',[]) @m5J%8>k
% axis square <~dfp
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h A ){>B<;
% end )3CM9P'0
% E.*hY+kGZ
% See also ZERNPOL, ZERNFUN2. SPV+ O{
edMCj
% Paul Fricker 11/13/2006 d7kE}{,
QKP
#wR
,?yjsJd.
% Check and prepare the inputs: wK2$hsque
% -----------------------------
:Hq%y/
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1vo3aF
error('zernfun:NMvectors','N and M must be vectors.') %O9 Wm_%
end ,&,%B|gT]
W$jRS
if length(n)~=length(m) ={xRNNUj_
error('zernfun:NMlength','N and M must be the same length.') D]E=0+
end DPf].i#
_qE9]mU
n = n(:); d[?RL&hJO
m = m(:); Yuv=<V
if any(mod(n-m,2)) igQzL*X
error('zernfun:NMmultiplesof2', ... "1gIR^S%9
'All N and M must differ by multiples of 2 (including 0).') 8d*S9p,/
end m u9,vH
V<WWtu;3
if any(m>n) drAJ-ii
error('zernfun:MlessthanN', ... DTC
IVLV
'Each M must be less than or equal to its corresponding N.') |vd|;" `
end B-LV/WJ_
$@6q5Iz!&
if any( r>1 | r<0 ) x9F* $G
error('zernfun:Rlessthan1','All R must be between 0 and 1.') yreH/$Ou8
end (^~a1@f,J
pbG-uH^
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j?C[ids<
error('zernfun:RTHvector','R and THETA must be vectors.') (tA[] ne2
end 7a_8007$l
VJ#ys_W
r = r(:); N=u(
3So
theta = theta(:); jy~hLEt7
length_r = length(r); cWnEp';.
if length_r~=length(theta) }'vQUGu8z
error('zernfun:RTHlength', ... 9=}#.W3.
'The number of R- and THETA-values must be equal.') 1;m?:|6K{
end \#biwX
~<u\YIJ
% Check normalization: d0T 8Cwcb
% -------------------- ?6*\M
if nargin==5 && ischar(nflag) 1g=T"O&=
isnorm = strcmpi(nflag,'norm'); b6;MTz*k>
if ~isnorm Ew]&~:$Ki
error('zernfun:normalization','Unrecognized normalization flag.') G-D}J2r=F
end X7*ossv
else WMKxGZg"
isnorm = false; I.t)sf,
end %l%ad-V
{6LS$3}VM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lrPIXIM
% Compute the Zernike Polynomials q !}~c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L|{v kkBo
I5j|\ /Ht
% Determine the required powers of r: 6GAEQ]
% ----------------------------------- }`Wo(E}O
m_abs = abs(m); TyO]|Q5
rpowers = []; D
Q4O
for j = 1:length(n) SIM>Lz
rpowers = [rpowers m_abs(j):2:n(j)]; F +5
5p8
end DQObHB8L
rpowers = unique(rpowers); xGOmvn^lQ
DQ$m@_/4w
% Pre-compute the values of r raised to the required powers, ~2[kCuu
% and compile them in a matrix: ??Urm[Y.Z
% ----------------------------- \Dr( /n
if rpowers(1)==0 v\(6uej^
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .5SYN-@
rpowern = cat(2,rpowern{:}); w}/+3z
rpowern = [ones(length_r,1) rpowern]; t0<RtIh9e
else ~"bBwPI
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z9MU%*N
rpowern = cat(2,rpowern{:}); M!hD`5.3
end ~mHrgxQ-
q(46v`u
% Compute the values of the polynomials: wk?i\vm
% -------------------------------------- (uHyWEHt
y = zeros(length_r,length(n)); e~he#o[%a
for j = 1:length(n) C!K&d,M
s = 0:(n(j)-m_abs(j))/2; sWTa;Qi
pows = n(j):-2:m_abs(j); +u |SX/C
for k = length(s):-1:1 x*j
eCD,
p = (1-2*mod(s(k),2))* ... a_VWgPVdDS
prod(2:(n(j)-s(k)))/ ... s,mt%^x[
prod(2:s(k))/ ... rw
2i_,.*~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Jzp|#*~$E
prod(2:((n(j)+m_abs(j))/2-s(k))); iu0'[
idx = (pows(k)==rpowers); vytO8m%U
y(:,j) = y(:,j) + p*rpowern(:,idx); L;Yn q<x
end h8k\~/iJ
.2|(!a9W
if isnorm UZ-pN_!Z:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3k8nWT:wT
end i$.! 8AV6
end av'[k<
% END: Compute the Zernike Polynomials -:!Wds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~USyN'5lU7
S%?%06$
% Compute the Zernike functions: J`; 9Z
% ------------------------------ (bpxj3@R
idx_pos = m>0; M)JozD%
idx_neg = m<0; {QN 5QGvK
XE0b9q954
z = y; +~==qLsU
if any(idx_pos) (B*,|D[J@i
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;B }4pv}
end z t
if any(idx_neg) 6\UIp#X
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -U7,~z
end |<8Fa%!HHc
wSZMHIW
% EOF zernfun