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 |SOLC
function z = zernfun(n,m,r,theta,nflag) Og 1-LP|X
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. KZ%i&w#<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N fbh,V%t7
% and angular frequency M, evaluated at positions (R,THETA) on the QCbD^
% unit circle. N is a vector of positive integers (including 0), and x-[ItJ% l
% M is a vector with the same number of elements as N. Each element Y1h)aQ5{
% k of M must be a positive integer, with possible values M(k) = -N(k) "Pwa}{
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, `6~0W5
% and THETA is a vector of angles. R and THETA must have the same ii?T:T@
% length. The output Z is a matrix with one column for every (N,M) HV~Fe!J_
% pair, and one row for every (R,THETA) pair. M8~3 0L
% 3=d%WPgQ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike uN)c!='I
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4
.
7X*1
% with delta(m,0) the Kronecker delta, is chosen so that the integral O^cC+@l!4
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s`$}xukT
% and theta=0 to theta=2*pi) is unity. For the non-normalized S"&Gutu3o
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KUJ Lx
% qx ki
% The Zernike functions are an orthogonal basis on the unit circle. EnWv9I<
% They are used in disciplines such as astronomy, optics, and EIRDH'[L
% optometry to describe functions on a circular domain. J1G}l5N
% UQu6JkbLL
% The following table lists the first 15 Zernike functions.
t1hQ0 B
% Vkg0C*L_
% n m Zernike function Normalization }<^mUG
% -------------------------------------------------- Eiu/p&ct
% 0 0 1 1 tu}!:5xi
% 1 1 r * cos(theta) 2 bny5e:= d
% 1 -1 r * sin(theta) 2 _Q1p_sdg
% 2 -2 r^2 * cos(2*theta) sqrt(6) k;^$Pd?t
% 2 0 (2*r^2 - 1) sqrt(3) 'W(u.
% 2 2 r^2 * sin(2*theta) sqrt(6) P*6m~`"5
% 3 -3 r^3 * cos(3*theta) sqrt(8) Z^> 4qf,k
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8)
!Vyf2xS"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iE'' >Z
% 3 3 r^3 * sin(3*theta) sqrt(8) 9qftMDLZJ\
% 4 -4 r^4 * cos(4*theta) sqrt(10) i M !`4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "s0,9;
}
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) UDJjw
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9E ^!i
% 4 4 r^4 * sin(4*theta) sqrt(10) 5!?5S$>
% -------------------------------------------------- I(*3n"
% E4% -*n
% Example 1: RH FRN&RU$
% gk|>E[.
% % Display the Zernike function Z(n=5,m=1) qKD
% x = -1:0.01:1; or*{P=m+R
% [X,Y] = meshgrid(x,x); jc"sPr v5
% [theta,r] = cart2pol(X,Y); 66&uK|
% idx = r<=1; 2jyWkAP'
% z = nan(size(X)); &<;T$Y
% z(idx) = zernfun(5,1,r(idx),theta(idx));
)c4tGT<
% figure 56)!&MF
% pcolor(x,x,z), shading interp B/;>v
% axis square, colorbar [_JdV(]$
% title('Zernike function Z_5^1(r,\theta)') `TPIc
% %4nf(|8n
% Example 2: |N`0G.#
% *,z/q6
% % Display the first 10 Zernike functions 4z(~)#'^
% x = -1:0.01:1;
b WNa6x
% [X,Y] = meshgrid(x,x); K[icVT2v~
% [theta,r] = cart2pol(X,Y); G*4I;'6
% idx = r<=1; *F1TZ_GS
% z = nan(size(X)); e8<}{N0,n
% n = [0 1 1 2 2 2 3 3 3 3]; Z4i))%or
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _]zX W
% Nplot = [4 10 12 16 18 20 22 24 26 28]; sMMOZ'bT
% y = zernfun(n,m,r(idx),theta(idx)); kf'(u..G
% figure('Units','normalized') v;\cM/&5
% for k = 1:10 "<=4]Z
% z(idx) = y(:,k); Ef`'r))
% subplot(4,7,Nplot(k)) W^8
% pcolor(x,x,z), shading interp Da 7(jA+
% set(gca,'XTick',[],'YTick',[]) TnN
ythwZ
% axis square KdkL_GSLT
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) w( V%EEk
% end 4*}&nmW
% S'!&,Dxq^
% See also ZERNPOL, ZERNFUN2.
oT\K P
/O:4u_
% Paul Fricker 11/13/2006 #$Zx ].[lc
L(yUS)O
u9 &$`N_G
% Check and prepare the inputs: "|X'qKS(H{
% ----------------------------- }B'-*)^|e{
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W+a/>U
error('zernfun:NMvectors','N and M must be vectors.') O5r8Ghf)
end '!^7 *@z
=Q<VU/
if length(n)~=length(m) vSHPN|*
error('zernfun:NMlength','N and M must be the same length.') H[hJUR+#
end 9>4 #I3
znE1t%V
n = n(:); p(pfJ^/:(
m = m(:); |^-D&C(Eu
if any(mod(n-m,2)) y!1X3X,V
error('zernfun:NMmultiplesof2', ... MU$tX
'All N and M must differ by multiples of 2 (including 0).')
ULt5Zi
end WkiT,(i
_]*YSeh=
if any(m>n) 4wSZ'RTSR
error('zernfun:MlessthanN', ... gfK_g)'2U
'Each M must be less than or equal to its corresponding N.') ow \EL
end U^KWRqt
_{`Z?lt
if any( r>1 | r<0 ) ;J|t-$Z
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 48wt
end h)Fc<,vwBE
{LjzkXs
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]<<,{IQ
error('zernfun:RTHvector','R and THETA must be vectors.') DyqqY$ vH(
end ;+\h$
#Gi`s?
r = r(:); !(q@sw(
theta = theta(:); 8$~oiK%fw
length_r = length(r); _p8u
&TZ
if length_r~=length(theta) ,+df=>$W
error('zernfun:RTHlength', ... !AXLoq$SY
'The number of R- and THETA-values must be equal.') xy:Mb =r
end b\JU%89
:oy2mi;
% Check normalization: r5xm7- `c
% -------------------- LC]0c)v#
if nargin==5 && ischar(nflag) BeFyx"NBg
isnorm = strcmpi(nflag,'norm'); J\@g3oGw
if ~isnorm bXJ(QXHd%
error('zernfun:normalization','Unrecognized normalization flag.') JL4E`
end bz>\n"'
else C')KZ|JIC
isnorm = false; ?<jWEz=
end lt-3OcC
Lx>[`QT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ez32k[eV!
% Compute the Zernike Polynomials ]0T*#U/P
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _yAY5TIv
B](R(x>L
% Determine the required powers of r: 9]+zZP_#
% ----------------------------------- _LZ(HTX~
m_abs = abs(m); OB9E30
rpowers = []; tRI<K
for j = 1:length(n) mTsyVji8
rpowers = [rpowers m_abs(j):2:n(j)]; gOnZ#
end Fk49~z
rpowers = unique(rpowers); G0!6rDu2,
0V-jOc
% Pre-compute the values of r raised to the required powers, Khd A;bF
% and compile them in a matrix:
}&+,y<>
% ----------------------------- #W8F_/!n|
if rpowers(1)==0 \xp0n
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !2Ompcr1
rpowern = cat(2,rpowern{:}); FR6 W-L
rpowern = [ones(length_r,1) rpowern]; .WKJ37od
else =c\(]xX
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \},H\kK+^
rpowern = cat(2,rpowern{:}); s:lH4B
end ^ U,iDK_
jY\z+lW6A
% Compute the values of the polynomials: g%=K
rO
% -------------------------------------- ].d%R a:{
y = zeros(length_r,length(n)); q}p$S2`
for j = 1:length(n) ShL!7y*rT{
s = 0:(n(j)-m_abs(j))/2; H.|I|XRG/
pows = n(j):-2:m_abs(j); G^ k8Or2
for k = length(s):-1:1 <gi~:%T
p = (1-2*mod(s(k),2))* ... ZRYlm$C
prod(2:(n(j)-s(k)))/ ... a$?d_BX
prod(2:s(k))/ ... hzk!H]>E
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .! <yTh
prod(2:((n(j)+m_abs(j))/2-s(k))); 9h+Hd&=
idx = (pows(k)==rpowers); ?J+jv
y(:,j) = y(:,j) + p*rpowern(:,idx); ::{\O\w
end '*XIp:
OcMB)1uh\
if isnorm | eCVq(R
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i 1w]j
end zd 2_k 9
end qJs_ahy(
% END: Compute the Zernike Polynomials @ NDcO,]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Ia'Yr
C3^3<
% Compute the Zernike functions: p=UW ^95
% ------------------------------ m$W2E.-$'#
idx_pos = m>0; _,0.h*c
idx_neg = m<0; Y(`Bc8h
qF^P\cD
z = y; O7IYg;
if any(idx_pos) >QJDO ]~V
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k(tB+k!vH\
end hd9~Zw]V
if any(idx_neg) 3/usgw1
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,F->*=
end 03)irq% l;
KM)MUPr
% EOF zernfun