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 ?!=yp#
function z = zernfun(n,m,r,theta,nflag) 095:"GvO
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K,{P
b?
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +G';no\h
% and angular frequency M, evaluated at positions (R,THETA) on the U}ei2q\
% unit circle. N is a vector of positive integers (including 0), and ]zEatY
% M is a vector with the same number of elements as N. Each element 45` i
% k of M must be a positive integer, with possible values M(k) = -N(k) .UF](
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \ s^a4l2
% and THETA is a vector of angles. R and THETA must have the same P` Hxj> {
% length. The output Z is a matrix with one column for every (N,M) '\8gY((7
% pair, and one row for every (R,THETA) pair. m~cz
% u`I&&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XKD0n^L[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7\>P@s
% with delta(m,0) the Kronecker delta, is chosen so that the integral U5N/'p%)<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (jbHV.]P9
% and theta=0 to theta=2*pi) is unity. For the non-normalized m2 0:{fld
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e P]L
% ru#,pJ=O(
% The Zernike functions are an orthogonal basis on the unit circle. NUBf>~_}
% They are used in disciplines such as astronomy, optics, and HriY-=ji>a
% optometry to describe functions on a circular domain. ~NT2QY5!K
% 5VD(fW[OW]
% The following table lists the first 15 Zernike functions. '4-J0S<<_
% f]Jn\7j4
% n m Zernike function Normalization \ng!qN
% -------------------------------------------------- nBw4YDR!
% 0 0 1 1 _L}k.
% 1 1 r * cos(theta) 2 Dv~W!T i
% 1 -1 r * sin(theta) 2 /J''`Tf
% 2 -2 r^2 * cos(2*theta) sqrt(6) -D*,*L
% 2 0 (2*r^2 - 1) sqrt(3) g\_J
% 2 2 r^2 * sin(2*theta) sqrt(6) WzD=Ol
% 3 -3 r^3 * cos(3*theta) sqrt(8) nn[OC=cDN
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) i\~@2
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MIa#\tJj
% 3 3 r^3 * sin(3*theta) sqrt(8) X{cFqW7
% 4 -4 r^4 * cos(4*theta) sqrt(10) J @eu]?h
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (QS4<J"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) .[mI9dc
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8@b,>l$
% 4 4 r^4 * sin(4*theta) sqrt(10) @JB9qT
% -------------------------------------------------- S7i,oP7
% F|mppY'<J
% Example 1: /e|vz^#+1,
% N_jpCCG~
% % Display the Zernike function Z(n=5,m=1) P){b"`f
% x = -1:0.01:1; D,R"P }G
% [X,Y] = meshgrid(x,x); l9Xz,H
% [theta,r] = cart2pol(X,Y); 1jHugss9|
% idx = r<=1; `Vph=`0
% z = nan(size(X)); 'uy\vR&Pz
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ]#$l"ss,
% figure f/"?(7F
% pcolor(x,x,z), shading interp i|N%dl+T=
% axis square, colorbar *vFXe_.
% title('Zernike function Z_5^1(r,\theta)') +95: O 8
% dgbqMu"
% Example 2: UdGa#rcNW
% 1u `{yl*+?
% % Display the first 10 Zernike functions $TU:iv1Fm
% x = -1:0.01:1; {&u`d.Lk2p
% [X,Y] = meshgrid(x,x); JSp V2c5Q
% [theta,r] = cart2pol(X,Y); A^L8"
% idx = r<=1; -_`dA^
% z = nan(size(X)); p. %lE!v
% n = [0 1 1 2 2 2 3 3 3 3]; @%"+;D
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; B}?$kp
% Nplot = [4 10 12 16 18 20 22 24 26 28]; FaA'%P@
% y = zernfun(n,m,r(idx),theta(idx)); ][D/=-
% figure('Units','normalized') F7!q18ew
% for k = 1:10 5~ip N/)E
% z(idx) = y(:,k); 77zfRSb+
% subplot(4,7,Nplot(k)) cc0e(\
% pcolor(x,x,z), shading interp GkU]>8E'"
% set(gca,'XTick',[],'YTick',[]) "pA24Ze
% axis square Zqi;by%
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Aq]*$s2\G
% end xZE%Gf_U
% ?z{Z!Bt?=)
% See also ZERNPOL, ZERNFUN2. zn+5pn&?
U"T>L
% Paul Fricker 11/13/2006 ,$oz1,Q/
sKCfI]
]ykMh
% Check and prepare the inputs: 9coN >y
% ----------------------------- sjW;Nsp
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'uBagd>*
error('zernfun:NMvectors','N and M must be vectors.') E9N.b.Q)
end !
<O,xI'
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dr/'
if length(n)~=length(m) (~U1X4
error('zernfun:NMlength','N and M must be the same length.') Y^(NzN
end nqv#?>Z^OT
yt5'2!jc
n = n(:); Yn>y1~
m = m(:); M/x*d4b_
if any(mod(n-m,2)) .ng:Z7
error('zernfun:NMmultiplesof2', ... ]"X} FU
'All N and M must differ by multiples of 2 (including 0).') nW"ml$
end 7dh--.i
1)N~0)dO
if any(m>n) b!l/O2
G
error('zernfun:MlessthanN', ... a?
<Ar#)j
'Each M must be less than or equal to its corresponding N.') 2;`F`}BA
end j0(+Kq:J
kN8?.V%Utw
if any( r>1 | r<0 ) ;p8,=w
error('zernfun:Rlessthan1','All R must be between 0 and 1.') nq,P.~l
end |]=s
tj<0q<is
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U/j+\Kc~
error('zernfun:RTHvector','R and THETA must be vectors.') ;)rs#T;$
end F#>^S9Gml
Q<UKR|6
r = r(:); iJ%`ym4Y
theta = theta(:); O8<@+xlX
length_r = length(r); ~'u %66
if length_r~=length(theta) #- z(]Y,y
error('zernfun:RTHlength', ... *#&s+h,^
'The number of R- and THETA-values must be equal.') Z.{r%W{2
end R2B0?fu
jHx)q|2\
% Check normalization: 1 GB
% -------------------- \CK f/:"
if nargin==5 && ischar(nflag) B`;DAsmT
isnorm = strcmpi(nflag,'norm'); <uL0M`u3
if ~isnorm $8t\|O3
error('zernfun:normalization','Unrecognized normalization flag.') ~'3hK4
end 0o*
else zrazFI0G
isnorm = false; ZnXq+^Z4
end RXWS,rF
38 HnW
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =k|hH~
% Compute the Zernike Polynomials (.J8Q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .:?cU#.
1z5\>F
% Determine the required powers of r: *s}j:fJ
% ----------------------------------- 7nOn^f D
m_abs = abs(m); )WR*8659e
rpowers = []; TkjPa};R
for j = 1:length(n) [R9!Tz
rpowers = [rpowers m_abs(j):2:n(j)]; ?[~)D}] j
end .!`v2_
rpowers = unique(rpowers); eK_Yt~dj
[-*8S1
% Pre-compute the values of r raised to the required powers, OK1f Y`$z
% and compile them in a matrix: %. -nZ C
% ----------------------------- ;x~[om21;
if rpowers(1)==0 l0g`;BI_
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /{7we$+,p
rpowern = cat(2,rpowern{:}); y|0I3n]e
rpowern = [ones(length_r,1) rpowern]; 8~s-@3J
else @[] A&)B
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); PdNxuy
rpowern = cat(2,rpowern{:}); f8X/kz
end eHy.<VX
M!E#T-)
% Compute the values of the polynomials: /naGn@m5u
% --------------------------------------
W;9Jah.
y = zeros(length_r,length(n)); dtT2h>h9
for j = 1:length(n) 8OW504AD
s = 0:(n(j)-m_abs(j))/2; | Sf` Cs
pows = n(j):-2:m_abs(j); A[.5Bi
for k = length(s):-1:1 va_TC!{;
p = (1-2*mod(s(k),2))* ... I-`qo7dQ_S
prod(2:(n(j)-s(k)))/ ... -a(\(^NW
prod(2:s(k))/ ... Y
=BXV7\
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *E- VS= #
prod(2:((n(j)+m_abs(j))/2-s(k))); fpK`
idx = (pows(k)==rpowers); +iL,8eW
y(:,j) = y(:,j) + p*rpowern(:,idx); HxmCKW!
end S3( 2.c~
!1M=9 ~$!
if isnorm T2$V5RyX
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $3C$])k
end D@yuldx'/
end b2vc
% END: Compute the Zernike Polynomials :%hxg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^MZdht
>&kb|)
% Compute the Zernike functions: `Wf)qMb
% ------------------------------ 0- 'f1 1S
idx_pos = m>0; U2(|/M+
idx_neg = m<0; |NiWr1&i0
T'TxC)
z = y; E*t0ia8
if any(idx_pos) U.@j!UrZ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); fDa$TbhjI
end t,8p}2,$
if any(idx_neg) #(`@D7S"
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y=6b oT
end .7nr :P
s: .5S
% EOF zernfun