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 0$P40 7
function z = zernfun(n,m,r,theta,nflag) RJGf@am&
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8mMrGf[Q\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ";xG[ne$Be
% and angular frequency M, evaluated at positions (R,THETA) on the Ot(EDa9}IJ
% unit circle. N is a vector of positive integers (including 0), and ofN|%g /
% M is a vector with the same number of elements as N. Each element 6KV&E8Gn
% k of M must be a positive integer, with possible values M(k) = -N(k) 4cs`R+]o
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, eyy&JjVs
% and THETA is a vector of angles. R and THETA must have the same gmrjCLj
% length. The output Z is a matrix with one column for every (N,M) /Bb\jvk-E
% pair, and one row for every (R,THETA) pair. /LJ?JwAvg5
% >yT:eG
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ww[STg
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <]"aP1+C
% with delta(m,0) the Kronecker delta, is chosen so that the integral Prr<:q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RMsr7M4<91
% and theta=0 to theta=2*pi) is unity. For the non-normalized :pOX,
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. x!Wl&
% F[Peil+|`
% The Zernike functions are an orthogonal basis on the unit circle. &?x^I{j
% They are used in disciplines such as astronomy, optics, and 20hE)!A
% optometry to describe functions on a circular domain. `kFxq<?aK
% qk<tLvD_'
% The following table lists the first 15 Zernike functions. ZLBfQ+pM)
% V_0e/7}Ya
% n m Zernike function Normalization "bC8/^
% -------------------------------------------------- Oq|pd7fcgm
% 0 0 1 1 }Z2Y>raA\
% 1 1 r * cos(theta) 2 gpO@xk$
% 1 -1 r * sin(theta) 2 |f`!{=?
% 2 -2 r^2 * cos(2*theta) sqrt(6) (swP#t5S
% 2 0 (2*r^2 - 1) sqrt(3) #{<Jm?sU
% 2 2 r^2 * sin(2*theta) sqrt(6) lQ)ZsFs=
% 3 -3 r^3 * cos(3*theta) sqrt(8) oA73\BFfP
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ynDa4HB
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8a"aJYj
% 3 3 r^3 * sin(3*theta) sqrt(8) oXfLNe6>L
% 4 -4 r^4 * cos(4*theta) sqrt(10) v%B^\S3)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) q"fK"H-j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $zDW)%nAX
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u5%.T0
P
% 4 4 r^4 * sin(4*theta) sqrt(10) Z @j0J[s
% -------------------------------------------------- {5_*tV<I
% K2)),_,@5+
% Example 1: G4ZeO:r
% l6a,:*_
% % Display the Zernike function Z(n=5,m=1) {8b6A~/
% x = -1:0.01:1; 6rdm=8WFA
% [X,Y] = meshgrid(x,x); `/0X].s#o
% [theta,r] = cart2pol(X,Y); .wYx_
% idx = r<=1; llQDZ}T
% z = nan(size(X)); YAd.i@^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); [ bE9Y;
% figure `W{Ye=|[d#
% pcolor(x,x,z), shading interp O{LWQ"@y
% axis square, colorbar L
+-B,466
% title('Zernike function Z_5^1(r,\theta)') 3u-j`7
% o^_z+JFwb
% Example 2: TQYud'u/
% 8h-6;x^^
% % Display the first 10 Zernike functions 9/q4]%`
% x = -1:0.01:1; kXv
-B-wOj
% [X,Y] = meshgrid(x,x); CEZ*a 0}=
% [theta,r] = cart2pol(X,Y); !P#lTyz
% idx = r<=1; A+:K!|w
% z = nan(size(X)); LV'v7 2yUH
% n = [0 1 1 2 2 2 3 3 3 3]; %xkqiI3Ff
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~99Ta]U
% Nplot = [4 10 12 16 18 20 22 24 26 28]; -KbT[]
% y = zernfun(n,m,r(idx),theta(idx)); 5-aCNAF2
% figure('Units','normalized') jbfMTb4
% for k = 1:10 =as ]>?<
% z(idx) = y(:,k); t$rWE|+_z
% subplot(4,7,Nplot(k)) 8[
:FU
% pcolor(x,x,z), shading interp p}O@%*p.
% set(gca,'XTick',[],'YTick',[]) 7$;mkHu4H%
% axis square ka*VQXk*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) X~%Wg*Hm
% end T?NwSxGo
% lv,8NmP5
% See also ZERNPOL, ZERNFUN2. vpTS>!i
]D%D:>9|/
% Paul Fricker 11/13/2006 ;. /Tv84I^
xOPSw|!w
&2#<6=}
% Check and prepare the inputs: JzCfs<D
% ----------------------------- !9OAMHa*9
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H&`p9d*(e
error('zernfun:NMvectors','N and M must be vectors.') K|E}Ni
end 9:4P7
2}'&38wMT
if length(n)~=length(m) Cm(Hu
error('zernfun:NMlength','N and M must be the same length.') ?cowey\m
.
end }=;N3Q" #y
%UY=VE\F
n = n(:); .KTDQA\
m = m(:); nEyPNm)
if any(mod(n-m,2)) 5|wQeosXxI
error('zernfun:NMmultiplesof2', ... c"77<Db$
'All N and M must differ by multiples of 2 (including 0).') pA"pt~6
end }a|SgI
[t,grdw
if any(m>n) FL"I PX;S
error('zernfun:MlessthanN', ... Fu!:8Wp!(
'Each M must be less than or equal to its corresponding N.') 5{[3I|m{
end Vr`UF0_3q
hFyN|Dqhds
if any( r>1 | r<0 ) @N1ta-D#
error('zernfun:Rlessthan1','All R must be between 0 and 1.') E}.cz\!.
end wW]|ElYR=
rXo,\zI;u^
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y-7x**I
error('zernfun:RTHvector','R and THETA must be vectors.') N{L ]H_=
end %[&cy'
nS]/=xP{
r = r(:); W;Ox H"eC
theta = theta(:); "?}QwtUW
length_r = length(r); A\.k['!
if length_r~=length(theta) ZLxe$.V_
error('zernfun:RTHlength', ... :G$NQ*(z
'The number of R- and THETA-values must be equal.') %t:1)]2
end &=K-~!?
%U-KQI0
% Check normalization: x!]ZVl]
% -------------------- jKM-(s!(
if nargin==5 && ischar(nflag) DM~Q+C=Yr
isnorm = strcmpi(nflag,'norm'); ezC55nm
if ~isnorm d cYUw]
error('zernfun:normalization','Unrecognized normalization flag.') RkP7}ZA;
end t.485L%
else d\'M ~VQ
isnorm = false; 0JKbp*H
end ]%"Z[R
_H<ur?G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W5EB+b49KM
% Compute the Zernike Polynomials C Vyq/X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `61VP-r
#oJ9BgDry
% Determine the required powers of r: 3Akb|r
% ----------------------------------- L}lc=\
m_abs = abs(m); /vwGSuk._
rpowers = []; J$]d%p_I
for j = 1:length(n)
=y[eQS$
rpowers = [rpowers m_abs(j):2:n(j)]; FwmE1,
end !N?|[n1
rpowers = unique(rpowers); .#lQZo6$\|
gj$gqO`B
% Pre-compute the values of r raised to the required powers, _+.z2} M
% and compile them in a matrix: *.ZV.(
% ----------------------------- &z&Jl#t-)
if rpowers(1)==0 D{PO!WzW
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9Z6O{
>
rpowern = cat(2,rpowern{:}); c
R[DT04
rpowern = [ones(length_r,1) rpowern]; CIYTs,u#
else 8{epy
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {*yhiE ,
rpowern = cat(2,rpowern{:}); wNcf7/ky
end q}1AV7$Ai
0_,V}
% Compute the values of the polynomials: Cp_"PvTmT
% -------------------------------------- E.}T.St
y = zeros(length_r,length(n)); L+9a4/q
for j = 1:length(n) "&77`R
s = 0:(n(j)-m_abs(j))/2; 7f~.Qus
pows = n(j):-2:m_abs(j); "Do9gW
for k = length(s):-1:1 rP^2MH"
p = (1-2*mod(s(k),2))* ... ceyZ4M
prod(2:(n(j)-s(k)))/ ... +'y$XR~W {
prod(2:s(k))/ ... W5HC7o\4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [gqV}Y"Md
prod(2:((n(j)+m_abs(j))/2-s(k))); jbMzcn~ehI
idx = (pows(k)==rpowers); (VU: &.
y(:,j) = y(:,j) + p*rpowern(:,idx); "qMd%RP
end u=p([
5]
sj0Hv d9
if isnorm {LrezE4
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u2@:[:Ao
end Ycn*aR2
end S^a")U4
% END: Compute the Zernike Polynomials Aum&U){yY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [;83
IoU}
#92MI#|n9
% Compute the Zernike functions: }9:d(B9;
% ------------------------------ gR?=z}`@p
idx_pos = m>0; 9p9:nx\
idx_neg = m<0; D)K/zh)
#zZQ@+5zw
z = y; H+;>>|+:~
if any(idx_pos) yAW%y
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3K_J"B*7
end m!tB;:6
if any(idx_neg) C8e{9CF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OU/PB
end o/)]z
z|<6y~5,
% EOF zernfun