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 ogtIn)
function z = zernfun(n,m,r,theta,nflag) HScj
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0dS}pd">k
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'J^ M`/
% and angular frequency M, evaluated at positions (R,THETA) on the E)==!T@E
% unit circle. N is a vector of positive integers (including 0), and GC?X>AC:
% M is a vector with the same number of elements as N. Each element [ZwZGAP
% k of M must be a positive integer, with possible values M(k) = -N(k) \zj _6Os
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, x._IP,vRx^
% and THETA is a vector of angles. R and THETA must have the same vZV+24YWb
% length. The output Z is a matrix with one column for every (N,M) a*LT <N
% pair, and one row for every (R,THETA) pair. u] C/RDTH
% A
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xlPUum-o
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Bvzu{B%
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0kN;SSX!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .C^1.)
% and theta=0 to theta=2*pi) is unity. For the non-normalized &gJKJ=7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,#3}TDC
% %bI(
% The Zernike functions are an orthogonal basis on the unit circle. .qVz rS
% They are used in disciplines such as astronomy, optics, and `5 py6,
% optometry to describe functions on a circular domain. Zgp]s+%E
% mv@cGdxu
% The following table lists the first 15 Zernike functions. ?pgdj|"a
% <hi@$.u_Q^
% n m Zernike function Normalization *8}Y0V\s
% -------------------------------------------------- nb(4"|8}
% 0 0 1 1 "|W .o=R
% 1 1 r * cos(theta) 2 K/RQ-xd4
% 1 -1 r * sin(theta) 2 PfX{n5yBW8
% 2 -2 r^2 * cos(2*theta) sqrt(6) X!5N2x
% 2 0 (2*r^2 - 1) sqrt(3) M=[ /v/M=
% 2 2 r^2 * sin(2*theta) sqrt(6) Q(SVJ
% 3 -3 r^3 * cos(3*theta) sqrt(8) c>fLSf
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) FFXDt"i2
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) YwGc[9=n
% 3 3 r^3 * sin(3*theta) sqrt(8) `x:znp} '
% 4 -4 r^4 * cos(4*theta) sqrt(10) v +-f
pl&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +3.Ik,Z}zq
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) }HS:3Dt
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )#-27Y
% 4 4 r^4 * sin(4*theta) sqrt(10) "sLdkd}dj
% -------------------------------------------------- T!$7:% D
% =jD[A>3I
% Example 1: h"VQFqQy
% )/k0*:OMyO
% % Display the Zernike function Z(n=5,m=1) Wz$%o'OnC
% x = -1:0.01:1; .4={K)kz|F
% [X,Y] = meshgrid(x,x); H e]1<tx
% [theta,r] = cart2pol(X,Y); `}o4 &$
% idx = r<=1; `NA[zH,w3
% z = nan(size(X)); G%)?jg@EA
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Wd4fIegk
% figure 7}bjJR "
% pcolor(x,x,z), shading interp GZT}aMMSJ
% axis square, colorbar K#M
h
% title('Zernike function Z_5^1(r,\theta)') /H.QGPr
% !8&,GT
% Example 2: ^|}C!t+
% k*|dX.C:
% % Display the first 10 Zernike functions
.fcU&t
% x = -1:0.01:1; }]vj"!?a
% [X,Y] = meshgrid(x,x); Okk[}G)
% [theta,r] = cart2pol(X,Y); 6QdNGpN
% idx = r<=1; WO*yJ`9]
% z = nan(size(X)); dsDoPo0!
% n = [0 1 1 2 2 2 3 3 3 3]; []Cvma1\
% m = [0 -1 1 -2 0 2 -3 -1 1 3];
7'FDI`e[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; "@B!5s0
% y = zernfun(n,m,r(idx),theta(idx)); z.16%@R
% figure('Units','normalized') vy/U""w`
% for k = 1:10 YVVX7hB
% z(idx) = y(:,k); R#~}ZUk2
% subplot(4,7,Nplot(k)) vZ
4Z+;.
% pcolor(x,x,z), shading interp c5P52_@
% set(gca,'XTick',[],'YTick',[]) i=_leC)rl
% axis square 7UHqiA`L
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $oE 4q6b
% end ^7q=E@[e
% *pP"u::S
% See also ZERNPOL, ZERNFUN2. nzy =0Ox[
&n<jpMB
% Paul Fricker 11/13/2006 5X&<+{bX
(Wr;:3i
zcJ]US
% Check and prepare the inputs: D{o1G?A
% ----------------------------- v,vTRrpK
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q" wi.&|
error('zernfun:NMvectors','N and M must be vectors.') mDE{s",q/
end Js+d4``W
w|WZEu:0|
if length(n)~=length(m) {+c/$4<
error('zernfun:NMlength','N and M must be the same length.') xmKa8']x
end qh$D;t1=
=khjD[muC
n = n(:); a2/r$Tgm
m = m(:); 4\pA^%73
if any(mod(n-m,2)) 7 g ]]>
error('zernfun:NMmultiplesof2', ... Z.6`O1OY}?
'All N and M must differ by multiples of 2 (including 0).') |UnTd$m
end P},S[GaZ
VK`_Qc#B
if any(m>n) uW>AH@Pij
error('zernfun:MlessthanN', ... p8s2#+/
'Each M must be less than or equal to its corresponding N.') xHsH .f_{
end mk\U wv
|+6Z+-.Hg
if any( r>1 | r<0 ) dMGu9k~u
error('zernfun:Rlessthan1','All R must be between 0 and 1.') fH`1dU
end 7P/j\frW
nWFp$tJ/R
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1)_f9GR
error('zernfun:RTHvector','R and THETA must be vectors.') ^\N2
Iu>6
end @mP@~
,_NO[+5U
r = r(:); #*S/Sh?Q
theta = theta(:); RB/[(4
length_r = length(r); *XH?|SV
if length_r~=length(theta) |D]jdd@!a2
error('zernfun:RTHlength', ... Jr17pu(t
'The number of R- and THETA-values must be equal.') aS~k.^N
end $#R.+B
w2{k0MW
% Check normalization: Jq6p5jr"
% -------------------- yWzvE:!)
if nargin==5 && ischar(nflag) u"T5m
isnorm = strcmpi(nflag,'norm'); PNc200`v4_
if ~isnorm ^|\ *i
error('zernfun:normalization','Unrecognized normalization flag.') 4]
?
end /SMp`Q88
else 8.-PQ
isnorm = false; -HoPECe
end pbqa
W@wT,yJ8@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ; UrwK
% Compute the Zernike Polynomials ,?&hqM\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8(3vNuyP
xmiF!R
% Determine the required powers of r: $6y1';A
% ----------------------------------- ;uoH+`pf
m_abs = abs(m); ][G<CO`k
rpowers = []; B/5C jHz
for j = 1:length(n) I*lq0&
rpowers = [rpowers m_abs(j):2:n(j)]; ~ S-x-cZ
end I5x/N.
rpowers = unique(rpowers); Y!POUMA
}A
?R,^prW{
% Pre-compute the values of r raised to the required powers, TqQ>\h"&_
% and compile them in a matrix: <