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 JHHb |
function z = zernfun(n,m,r,theta,nflag) n&3iz05}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. aS2a_!f
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rE9Ta8j6
% and angular frequency M, evaluated at positions (R,THETA) on the
uT#Acg
% unit circle. N is a vector of positive integers (including 0), and iz,]%<_PE
% M is a vector with the same number of elements as N. Each element 5^bh.uF
% k of M must be a positive integer, with possible values M(k) = -N(k) 7O]J^H+7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Bi %Z2/
% and THETA is a vector of angles. R and THETA must have the same !>?4[|?n<
% length. The output Z is a matrix with one column for every (N,M) ccIDMJ=2
% pair, and one row for every (R,THETA) pair. `4se7{'UK`
% eUi> Mp
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike NU BpIx&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z&\Il#'\m+
% with delta(m,0) the Kronecker delta, is chosen so that the integral nYo&x'
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hqdC9?\
% and theta=0 to theta=2*pi) is unity. For the non-normalized 721{Ga4~S
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9<+;hH8J_r
% 7g {g}
% The Zernike functions are an orthogonal basis on the unit circle. y^5T/M
% They are used in disciplines such as astronomy, optics, and 8') .ohD
% optometry to describe functions on a circular domain. U]+b`m
% `M towXj
% The following table lists the first 15 Zernike functions. #i'C
% 7[(Lrx.pM
% n m Zernike function Normalization _Ac/i r[,:
% -------------------------------------------------- 7*R{u*/e
% 0 0 1 1 !3O,DhH>MC
% 1 1 r * cos(theta) 2 ){?mKB5
% 1 -1 r * sin(theta) 2 m~A[V,os
% 2 -2 r^2 * cos(2*theta) sqrt(6) Nv}U/$$S
% 2 0 (2*r^2 - 1) sqrt(3) V'Sd[*
% 2 2 r^2 * sin(2*theta) sqrt(6) TyxU6<>4J4
% 3 -3 r^3 * cos(3*theta) sqrt(8) \SoYx5lf
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) m70`{-O
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ^K1~eb*K
% 3 3 r^3 * sin(3*theta) sqrt(8) xkk@{}J\
% 4 -4 r^4 * cos(4*theta) sqrt(10) N>W;0u!
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G_4K+
-K
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [u!p-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]j%*"V
% 4 4 r^4 * sin(4*theta) sqrt(10) A52LH,
% -------------------------------------------------- 2tg/S=t}
% E7d~#
% Example 1: AQJ|^'%
% ^=4I|+P,6.
% % Display the Zernike function Z(n=5,m=1) Huc3|~9
% x = -1:0.01:1; u&?yPR
% [X,Y] = meshgrid(x,x); !;xf>API
% [theta,r] = cart2pol(X,Y); Zi2Eu4p l{
% idx = r<=1; Mm:a+T
% z = nan(size(X)); E-5ij,bHv3
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Qd&d\w/
% figure \UkNE5
% pcolor(x,x,z), shading interp e{qp!N1!
% axis square, colorbar Xy3g(x]
% title('Zernike function Z_5^1(r,\theta)') qY*%p
% 46Y7HTwE
% Example 2: 8o%<.]
% V{a}#J
% % Display the first 10 Zernike functions 2yi*eR
% x = -1:0.01:1;
]*kP>
% [X,Y] = meshgrid(x,x); mlsvP%[f.
% [theta,r] = cart2pol(X,Y); #2ZrdD"5kQ
% idx = r<=1; ~x+:44*
% z = nan(size(X)); L:k@BCQM
% n = [0 1 1 2 2 2 3 3 3 3]; HzgQI
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; rS,*s'G
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 4X(1
% y = zernfun(n,m,r(idx),theta(idx)); j:de}!wc
% figure('Units','normalized') ~8Dd<4?F]
% for k = 1:10 z Et6
% z(idx) = y(:,k); ~]6Oz;~<3
% subplot(4,7,Nplot(k)) U:etcnb4w>
% pcolor(x,x,z), shading interp ]`CKQ>
o
% set(gca,'XTick',[],'YTick',[]) 5sA>O2Rt>
% axis square I49=ozPP
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) SoM
]2^
% end y$r?t0
% btB(n<G2#
% See also ZERNPOL, ZERNFUN2. n'x`oI)-
7DHT)9lD/
% Paul Fricker 11/13/2006 zn?a|kt
{8>_,z^P)
JJbM)B@-
% Check and prepare the inputs: h!t2H6eyF
% ----------------------------- .eDxIWW+ft
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /FNj|7s
error('zernfun:NMvectors','N and M must be vectors.') Tg{dIh.Q~O
end !,-qn)b
u6bB5(s`&
if length(n)~=length(m) o}AqNw60v
error('zernfun:NMlength','N and M must be the same length.') dTU.XgX)1^
end 4o)\DB?!
zM9) .D
H
n = n(:); I;|5C=!
m = m(:); Sj]T{3mi
if any(mod(n-m,2)) 40l#'< y;
error('zernfun:NMmultiplesof2', ... yrK--C8
'All N and M must differ by multiples of 2 (including 0).') Ik@Q@ T"
end "#eNFCo7k
Jj^<:t5{rN
if any(m>n) 5sV/N] !
error('zernfun:MlessthanN', ... Ph7(JV{
'Each M must be less than or equal to its corresponding N.') T$8$9D_u
end "`1of8$X7
e&a[k
if any( r>1 | r<0 ) [2H(yLw O
error('zernfun:Rlessthan1','All R must be between 0 and 1.') WHD/s
end [0,q7d?"
#*;fQ&p
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
"ppb%=
error('zernfun:RTHvector','R and THETA must be vectors.') c_8 mQ
end $0`$)(Y
7yCx !P;
r = r(:); qwq+?fj={
theta = theta(:); JXR/K=<^
length_r = length(r); G~$M"@Q7N
if length_r~=length(theta) ]@<3 6ByM
error('zernfun:RTHlength', ... !A^w6Q;`V
'The number of R- and THETA-values must be equal.') ?PxYS%D_L
end %H 6ZfEO
IkXKt8`YVA
% Check normalization: .1? i'8TF
% -------------------- aBtfZDCfzp
if nargin==5 && ischar(nflag) /Geks/
isnorm = strcmpi(nflag,'norm'); TAXkfj
if ~isnorm ([XyW{=h!
error('zernfun:normalization','Unrecognized normalization flag.') z&yb_A:>
end p$!+2=)gY
else DSG +TA"
isnorm = false; fM[fS?W
end Qc
=lf$
17[t_T&Ak9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &+r
;>
% Compute the Zernike Polynomials Px?At5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >zx50e)
[F-u'h< *l
% Determine the required powers of r: g}og@UY7#
% ----------------------------------- =`.5b:e
m_abs = abs(m); t:j07 ,1~
rpowers = []; ^)P5(fJ
for j = 1:length(n) 9qO:K79|
rpowers = [rpowers m_abs(j):2:n(j)]; K}*p(1$u
end 1X_!%Z
rpowers = unique(rpowers); U!UX"r
H=SMDj)s+
% Pre-compute the values of r raised to the required powers, VS@W.0/
% and compile them in a matrix: ZYt"=\_
% ----------------------------- d~bH!P
if rpowers(1)==0 ^A$XXH'
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -clg'Aa;.
rpowern = cat(2,rpowern{:}); G;#t6bk
rpowern = [ones(length_r,1) rpowern]; jE5
9h
else ~Wd8>a{w
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5}^08Xl
rpowern = cat(2,rpowern{:}); !";$Zu
end 8\t7}8f
H.G^!0j;
% Compute the values of the polynomials: \c^jaK5
% -------------------------------------- $A0]v!P~i-
y = zeros(length_r,length(n)); |q b92|?
for j = 1:length(n) k)t8J \
s = 0:(n(j)-m_abs(j))/2; 7}7C0mV3
pows = n(j):-2:m_abs(j); -#z'A
for k = length(s):-1:1 G/;aZ
p = (1-2*mod(s(k),2))* ... 91Sb=9
prod(2:(n(j)-s(k)))/ ... 0_Z|y/I.
prod(2:s(k))/ ... <T~fh>a
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ZaV66Y>
prod(2:((n(j)+m_abs(j))/2-s(k))); ?U[nYp}"v
idx = (pows(k)==rpowers); ~=]@],{
y(:,j) = y(:,j) + p*rpowern(:,idx); Gkvd{G?F
end _[Wrd?Z
3T^dgWXEG
if isnorm >!.lr9(l
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); !x /Z"
end +GtGyp
end %SFR.U0}yK
% END: Compute the Zernike Polynomials -.3k
vL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g5N<B+?!i
Q"_T040B
% Compute the Zernike functions: Y-k~ 7{7
% ------------------------------ f;dU72]q+
idx_pos = m>0; gx
R|S
idx_neg = m<0; d(tf: @
WC; a
z = y; ON!G{=7
if any(idx_pos) jJC((1|
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #mxfU>vQ:
end F09AX'nj
if any(idx_neg) Eu~wbU"%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); q)y8Bv|
end P&,cCR>
|W];v@b\y
% EOF zernfun