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 xfjd5J7'
function z = zernfun(n,m,r,theta,nflag) ^+ZgWS^%
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |%
z^N*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !p9)CjQ "
% and angular frequency M, evaluated at positions (R,THETA) on the ! Tx&vtq
% unit circle. N is a vector of positive integers (including 0), and 96d~~2p
% M is a vector with the same number of elements as N. Each element HcRa`Sfc]/
% k of M must be a positive integer, with possible values M(k) = -N(k) JVtQ,oZ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, *5_V*v6
% and THETA is a vector of angles. R and THETA must have the same QK)){cK
% length. The output Z is a matrix with one column for every (N,M) pkJ/oT
% pair, and one row for every (R,THETA) pair. R}8XRe
% v??TJ^1
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike u*3NS$vH
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 0RUi\X4HI
% with delta(m,0) the Kronecker delta, is chosen so that the integral ~#R9i^Y
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x2co>.i
% and theta=0 to theta=2*pi) is unity. For the non-normalized WJ|:kuF
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. rcV-_+KE(B
% ^$v3eKA
% The Zernike functions are an orthogonal basis on the unit circle. n]Zk;%yL
% They are used in disciplines such as astronomy, optics, and e,>%Z@92(
% optometry to describe functions on a circular domain. NYwR2oX
% ~@T<gA9V
% The following table lists the first 15 Zernike functions. >nzu],U
% M|q~6oM
% n m Zernike function Normalization *O,H5lwU
% -------------------------------------------------- 41G5!=i
% 0 0 1 1 O.,3|
% 1 1 r * cos(theta) 2 7FLXx?nLY
% 1 -1 r * sin(theta) 2 rq sdE
% 2 -2 r^2 * cos(2*theta) sqrt(6) "g>.{E5
% 2 0 (2*r^2 - 1) sqrt(3) G?AG:%H %
% 2 2 r^2 * sin(2*theta) sqrt(6) fmfTSN(Q~`
% 3 -3 r^3 * cos(3*theta) sqrt(8) {ox2Tg?
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) K{@3\5<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .Da'pOe
% 3 3 r^3 * sin(3*theta) sqrt(8) :w`3cwQ
% 4 -4 r^4 * cos(4*theta) sqrt(10) (-0ePSOG
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?-MP_9!JK
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 20b<68h$:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >G~mp<L
% 4 4 r^4 * sin(4*theta) sqrt(10) BecPT
% -------------------------------------------------- LJFG0 W
% n(1')?"mA
% Example 1: (@r
`$5D.b
% #*9-d/K
% % Display the Zernike function Z(n=5,m=1) .B72C[' c
% x = -1:0.01:1; `Out(Hn
% [X,Y] = meshgrid(x,x); 3*ixlO:qGk
% [theta,r] = cart2pol(X,Y); POAw M
% idx = r<=1; U!(@q!>G
% z = nan(size(X)); vAb^]d
% z(idx) = zernfun(5,1,r(idx),theta(idx)); SJ?6{2^
% figure 7%MbhlN.
% pcolor(x,x,z), shading interp X(A.X:"
% axis square, colorbar (xl\J/
% title('Zernike function Z_5^1(r,\theta)') #m<tJnEO
% GsQ*4=C
% Example 2: KS}hU~
% 31WC=ur5
% % Display the first 10 Zernike functions @{hd{>K*
% x = -1:0.01:1; q%(EYM5Y
% [X,Y] = meshgrid(x,x); C NsNZJ
% [theta,r] = cart2pol(X,Y); @I`C#~
% idx = r<=1; urBc=3Rz
% z = nan(size(X)); vb
Y3;+M>
% n = [0 1 1 2 2 2 3 3 3 3]; 0'5/K ,
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;G |i^
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ;5_{MCPM
% y = zernfun(n,m,r(idx),theta(idx)); t5B7I59
% figure('Units','normalized') <TGn=>u
% for k = 1:10 hR#-u1C
% z(idx) = y(:,k); e~l#4{w
% subplot(4,7,Nplot(k))
h `}}
% pcolor(x,x,z), shading interp VU`OO$,W
% set(gca,'XTick',[],'YTick',[]) oA] KE"T
% axis square sRSz}]
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7hP<f}xL
% end k%s_0
@
% =m89z}Ot
% See also ZERNPOL, ZERNFUN2. #Z+i~t{e(
r;BT,jiX
% Paul Fricker 11/13/2006 ~{hxR)x9
az0<5Bq)
Fm\"{)V:b
% Check and prepare the inputs: +4;uF]T
% ----------------------------- ; Uc0o!1
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v
^[39*8
error('zernfun:NMvectors','N and M must be vectors.') YHNR3
end 2H71~~ c
!oPq?lW9
if length(n)~=length(m) Hnknly
error('zernfun:NMlength','N and M must be the same length.') q<y#pL=k"*
end SIO&rrT.
o#) {1<0vg
n = n(:); 'c2W}$q
m = m(:); **9x?s
if any(mod(n-m,2)) :NJ_n6E
error('zernfun:NMmultiplesof2', ... ]]7mlQ
'All N and M must differ by multiples of 2 (including 0).') j',W 64
end vgY3L
W} WI; cI
if any(m>n) {3;AwhN0H
error('zernfun:MlessthanN', ... `&\Q +W
'Each M must be less than or equal to its corresponding N.') T134ZXqqz
end 8fA_p}wp
Vk<
LJ
S
if any( r>1 | r<0 ) KT]Pw\y5
error('zernfun:Rlessthan1','All R must be between 0 and 1.') D\IjyZ-O
end Uc/+gz
Z;
4tL<q_
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _zlqtO
error('zernfun:RTHvector','R and THETA must be vectors.') J+rCxn?;g
end F,
U*yj
l/;X?g5+
r = r(:); %ZHP2j
%~
theta = theta(:); UOQEk22
length_r = length(r); ;iDPn2?6?x
if length_r~=length(theta) zJe#m|Z
error('zernfun:RTHlength', ... r0p w_j
'The number of R- and THETA-values must be equal.') d%l{V6
end %%(R@kh9
wFG3KzEq ~
% Check normalization: {U&.D
[{&
% -------------------- rG,5[/l
if nargin==5 && ischar(nflag) V_plq6z
isnorm = strcmpi(nflag,'norm'); IV\J3N^
if ~isnorm ]Q[p@gLd
error('zernfun:normalization','Unrecognized normalization flag.') U,nEbKJgk
end GfM;saTz{
else 'SQG>F Uy
isnorm = false; hiNEJ_f
end l5L.5$N
!i=nSqW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VfT*7_
% Compute the Zernike Polynomials xf|mlHS+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [+qCs7'
bn
|zl!Pq
% Determine the required powers of r: Da"j E
% -----------------------------------
}fp-5
m_abs = abs(m); ,3nN[)dk
rpowers = []; 2<M= L1\
for j = 1:length(n) 9"g6C<
rpowers = [rpowers m_abs(j):2:n(j)]; ?%H):r
end iNMx"F0r
rpowers = unique(rpowers); Tw +
Nk {XdrY
% Pre-compute the values of r raised to the required powers, {BKl` 1z
% and compile them in a matrix: odIZo|dv
% ----------------------------- GR\5WypoJ
if rpowers(1)==0 S_~z-`;h!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LM2TZ
rpowern = cat(2,rpowern{:}); @LJpdvb
rpowern = [ones(length_r,1) rpowern]; 610D%F
else M DF%\Sx
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); bXS:x
rpowern = cat(2,rpowern{:}); !UFfsNiXZ
end z0/}
!
9c JH"
% Compute the values of the polynomials: 5xii(\lC
% -------------------------------------- u, 3#M ~
y = zeros(length_r,length(n)); .!JVr"8
for j = 1:length(n) PfkrOsV/m
s = 0:(n(j)-m_abs(j))/2; Y#g4$"G9
pows = n(j):-2:m_abs(j); 7'OtruJ
for k = length(s):-1:1 !0N7^Z"gtz
p = (1-2*mod(s(k),2))* ... <})'Y~i
prod(2:(n(j)-s(k)))/ ... 6m6zA/
prod(2:s(k))/ ... qc-mGmom L
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IgC}&
prod(2:((n(j)+m_abs(j))/2-s(k))); ^B+!N;
idx = (pows(k)==rpowers); -,["c9'3
y(:,j) = y(:,j) + p*rpowern(:,idx); j;+?HbL
end SXt{k<|
Z{H5oUk
if isnorm A'nq}t 3
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v!%5&: c3
end 8XsguC
end ^Idle*+
% END: Compute the Zernike Polynomials Vx @|O%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $y
b4xU
g(#f:"
% Compute the Zernike functions: [V}S<Xp
% ------------------------------ . BiCBp<
idx_pos = m>0; uPniLx\t:
idx_neg = m<0; &7_Qd4=08w
T6~_Q}6
z = y; UQ4% Xp
if any(idx_pos) Pzb|t+"$
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Rar"B*b;$
end +kFxi2L6
if any(idx_neg) ,~?YBLw@c
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .$#rV?7
end Dr6A,3B
8|$3OVS
% EOF zernfun