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 niCK(&z
function z = zernfun(n,m,r,theta,nflag) 'ux!:b"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. O'IU1sU
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mST8+R@S
% and angular frequency M, evaluated at positions (R,THETA) on the s&pnB
% unit circle. N is a vector of positive integers (including 0), and }\S'oC\[
% M is a vector with the same number of elements as N. Each element Cp/f18zO
% k of M must be a positive integer, with possible values M(k) = -N(k) Uc:NW
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ~IW{^u
% and THETA is a vector of angles. R and THETA must have the same O<Q8%Az
% length. The output Z is a matrix with one column for every (N,M) b4dviYI
% pair, and one row for every (R,THETA) pair. 8Yk*$RR9
% .B<Bqr@?8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Dq~;h \='
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), pD({"A.x9z
% with delta(m,0) the Kronecker delta, is chosen so that the integral X-nC2[tu'W
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W;=Ae~
% and theta=0 to theta=2*pi) is unity. For the non-normalized l+ >eb
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. XfE9QA[
% 1D#-,#?
% The Zernike functions are an orthogonal basis on the unit circle. JqMF9|{H
% They are used in disciplines such as astronomy, optics, and .e0)@}Jv8>
% optometry to describe functions on a circular domain. TMMJ5\t2
% |?VJf3A
% The following table lists the first 15 Zernike functions. p&RC#wYu
% B%uY/Mwz$
% n m Zernike function Normalization -O\i^?lD;
% -------------------------------------------------- HdxP:s.T
% 0 0 1 1 'o}[9ZBjn
% 1 1 r * cos(theta) 2 MAkr9AKb,
% 1 -1 r * sin(theta) 2 ;DK%!."%
% 2 -2 r^2 * cos(2*theta) sqrt(6) K [DpH&
% 2 0 (2*r^2 - 1) sqrt(3) }r@dZBp:
% 2 2 r^2 * sin(2*theta) sqrt(6) &
V>rq'~;
% 3 -3 r^3 * cos(3*theta) sqrt(8) y&
yf&p
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) V($V8P/
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Godrz*"
% 3 3 r^3 * sin(3*theta) sqrt(8) #PD6LO
% 4 -4 r^4 * cos(4*theta) sqrt(10) gm)Uyr$
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LE<J<~2Z
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) M]r?m@)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;_"|#
% 4 4 r^4 * sin(4*theta) sqrt(10) ,9bnR;f\
% -------------------------------------------------- FiiDmhu
% o:Kw<z,$H
% Example 1: U&WEe`XM
% Kb(11$U
% % Display the Zernike function Z(n=5,m=1) b*?u+tWP_
% x = -1:0.01:1; =D$ED^W
% [X,Y] = meshgrid(x,x); t([}a~1}
% [theta,r] = cart2pol(X,Y); !-7n69:G
% idx = r<=1; @p*)^D6E\
% z = nan(size(X)); Zw9;g+9
% z(idx) = zernfun(5,1,r(idx),theta(idx)); * $f`ouJl
% figure lcZ.}
% pcolor(x,x,z), shading interp I2*rtVAP'j
% axis square, colorbar &t9V
% title('Zernike function Z_5^1(r,\theta)') yV8J-YdsG
% RN(I}]] a
% Example 2: _aPAn|.
% ;`#R9\C=h
% % Display the first 10 Zernike functions A!bG 2{r
% x = -1:0.01:1; /dYv@OU?
% [X,Y] = meshgrid(x,x); VdK%m`;2
% [theta,r] = cart2pol(X,Y); 3>1^$0iq
% idx = r<=1; W\kli';jyC
% z = nan(size(X)); kh0cJE\_^
% n = [0 1 1 2 2 2 3 3 3 3]; EB*sd S
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; f zo'9
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Os"('@jd>
% y = zernfun(n,m,r(idx),theta(idx)); ^-Od*DTL
% figure('Units','normalized') r+FEgSDa]
% for k = 1:10 [HQ)4xG
% z(idx) = y(:,k); 3{3@>8{w
% subplot(4,7,Nplot(k)) w95M
B*N
% pcolor(x,x,z), shading interp }'x;J
% set(gca,'XTick',[],'YTick',[]) \2s`mCY
% axis square _Ub
`\ytx
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) hN'])[+V
% end pIlEoG=[_
% (P)G|2=
% See also ZERNPOL, ZERNFUN2. . ImaM
5X!-Hj
% Paul Fricker 11/13/2006 _!',%+
-)}s{[]d6m
nzflUR{`-
% Check and prepare the inputs: )Zr9
`3[
% ----------------------------- '}_r/l]K
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -x//@8"
error('zernfun:NMvectors','N and M must be vectors.') }sXTZX
end 1]7gYNzV"
_B^zm-}8|B
if length(n)~=length(m) n"EKVw7Y
error('zernfun:NMlength','N and M must be the same length.') $6"(t= %{
end F^O83[S
~gfR1SE
n = n(:); qE~_}4\Z9
m = m(:); hN-@_XSw<I
if any(mod(n-m,2)) hk~/W}sI
error('zernfun:NMmultiplesof2', ... )Z/"P\qo
'All N and M must differ by multiples of 2 (including 0).') "bo0O7InOV
end P"w\hF
Rg?6e N
if any(m>n) Z4] n<~o
error('zernfun:MlessthanN', ... P3_.U8g$r
'Each M must be less than or equal to its corresponding N.') <sH}X$/
end \Rny*px
L80(9Y^xn
if any( r>1 | r<0 ) ?"d$SK"6Z
error('zernfun:Rlessthan1','All R must be between 0 and 1.') fW-C`x
end t7+A!7b{
q\Y4v Wg
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z]G|)16
error('zernfun:RTHvector','R and THETA must be vectors.') kU<t~+
end iEvQ4S6tD
tq3_az ~1
r = r(:); V_+&Y$msi~
theta = theta(:); ^dQ{vL@9b9
length_r = length(r); 4V,.Oi
if length_r~=length(theta) .Nn11F< d
error('zernfun:RTHlength', ... 4yl{:!la
'The number of R- and THETA-values must be equal.') ffrIi',@
end _[2@2q0
":Wq<Z'
% Check normalization: bNea5u##
% -------------------- Y?0/f[Ax,y
if nargin==5 && ischar(nflag) JVE\{ e)
isnorm = strcmpi(nflag,'norm'); GShxPH{_j
if ~isnorm j_Szw
w-
error('zernfun:normalization','Unrecognized normalization flag.') %**f`L%jN
end P-@MLIC{
else [^5\Ww
isnorm = false; =S&`~+
end j6rN t|
6}4})B2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QU).q65p
% Compute the Zernike Polynomials 4qQ,1&!]S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P49\A^5S!
3A7774n=P
% Determine the required powers of r: :L[>!~YG_n
% ----------------------------------- D|;O9iks#
m_abs = abs(m); r"7n2
rpowers = []; #.Rn6|V/4
for j = 1:length(n) sXIYl% d
rpowers = [rpowers m_abs(j):2:n(j)]; </h^%mnd
end V>{< pS
rpowers = unique(rpowers); h@:K=ggK
8H!QekQZ]\
% Pre-compute the values of r raised to the required powers, 9j,g&G.K
% and compile them in a matrix: z|l*5@p
% ----------------------------- Ni,nQ;9
if rpowers(1)==0 c`a(
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R@vcS=m7
rpowern = cat(2,rpowern{:}); %Sr+D{B
rpowern = [ones(length_r,1) rpowern]; V`V\/s gj
else Z~5) )5Ye;
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); tdy2ZPVtTV
rpowern = cat(2,rpowern{:}); *IG$"nu
end ?e7]U*jEU
^t;z;.g
% Compute the values of the polynomials: r~4uIUE{
% -------------------------------------- J$dwy$n
y = zeros(length_r,length(n)); IrLGAQ0
for j = 1:length(n) rwm^{Qa
s = 0:(n(j)-m_abs(j))/2; C-'hXh;hQ
pows = n(j):-2:m_abs(j); }lJ;|kx$
for k = length(s):-1:1 }cKB)N
BJb
p = (1-2*mod(s(k),2))* ... ?^}30V:E
prod(2:(n(j)-s(k)))/ ... U.%Kt,qB
prod(2:s(k))/ ... {z#2gc'Q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... rqdwQ
prod(2:((n(j)+m_abs(j))/2-s(k))); o2 14V \
idx = (pows(k)==rpowers); |c_qq Bd
y(:,j) = y(:,j) + p*rpowern(:,idx); V~J5x >O
end K=g</@L6R
()3\(d5e
if isnorm x%{]'z
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (\V
i_
end bOS)vt*V
end c0!.ei
% END: Compute the Zernike Polynomials op,L3:R\Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M>8J_{r^
M6Fo.eeK3
% Compute the Zernike functions: em$pU*`P
% ------------------------------ 7R+(3NU1A
idx_pos = m>0; -%K!Ra\W
idx_neg = m<0; g?C;b>4
AOf4y&B>q
z = y; VFHd2Ea(
if any(idx_pos) 39pG-otJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *{o7G a
end SC{m@
if any(idx_neg) hlTbCl
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6_LeP9s )
end e|~MJu+1
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% EOF zernfun