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 RRK^~JQI.2
function z = zernfun(n,m,r,theta,nflag) 1v+JCOy
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `kI?Af*;v
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 56/.*qa
% and angular frequency M, evaluated at positions (R,THETA) on the |E>v~qD8I
% unit circle. N is a vector of positive integers (including 0), and UXXqE4x
% M is a vector with the same number of elements as N. Each element vy>];!Cu
% k of M must be a positive integer, with possible values M(k) = -N(k) eG a#$x?.
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \3J+OY
% and THETA is a vector of angles. R and THETA must have the same Y0R\u\b
% length. The output Z is a matrix with one column for every (N,M) P*?d6v,r
% pair, and one row for every (R,THETA) pair. x0N-[//YV
% E,"b*l.
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /S-/SF:>g
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), k:&?$
% with delta(m,0) the Kronecker delta, is chosen so that the integral hnM9-hqm
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7=9A_4G!
% and theta=0 to theta=2*pi) is unity. For the non-normalized HY@kw>I
% polynomials, max(Znm(r=1,theta))=1 for all [n,m].
Ep#<$6>
% HXlr
% The Zernike functions are an orthogonal basis on the unit circle. G~Q*:m
% They are used in disciplines such as astronomy, optics, and bqf]$}/8k
% optometry to describe functions on a circular domain. 4okHAv8;
% , 4h!"c
% The following table lists the first 15 Zernike functions. R(n0!h4
% v ](G?L9b
% n m Zernike function Normalization ,Yiq$Z{qQ
% -------------------------------------------------- #]N&6ngJ
% 0 0 1 1 K{`2jK#
% 1 1 r * cos(theta) 2 o{YW
% 1 -1 r * sin(theta) 2 {!:|.!-u
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?[*@T2Ck
% 2 0 (2*r^2 - 1) sqrt(3) J"a2
@S&
% 2 2 r^2 * sin(2*theta) sqrt(6) Xm0&U?dZB
% 3 -3 r^3 * cos(3*theta) sqrt(8) NUxAv= xl
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) wUZ(Tin
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iPtm@f,bI
% 3 3 r^3 * sin(3*theta) sqrt(8) !Ed<xG/
% 4 -4 r^4 * cos(4*theta) sqrt(10) iYmzk?U
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {U+9,6.`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ?()E5 4y
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "=v J}
% 4 4 r^4 * sin(4*theta) sqrt(10) :*w:eKk
% -------------------------------------------------- (pRy1DH~
% 0N}
wD-
% Example 1: " N`V*0h
% o+6^|RP
% % Display the Zernike function Z(n=5,m=1) [4+a 1/^
% x = -1:0.01:1; s K$Sar
% [X,Y] = meshgrid(x,x); eL]w' }\
% [theta,r] = cart2pol(X,Y); =":V
WHf
% idx = r<=1; k*UR#z(I
% z = nan(size(X)); ^0,&R\e+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p1`'1`.3
% figure W0r5D9k
% pcolor(x,x,z), shading interp aS1P]&
% axis square, colorbar (fLbg,
% title('Zernike function Z_5^1(r,\theta)') Hhce:E@K
% ko7-%+0|]
% Example 2: Ow&'sR'CX
% ?-6x]l=]
% % Display the first 10 Zernike functions 0I
ND9h.%
% x = -1:0.01:1; BR0p0%
% [X,Y] = meshgrid(x,x); szM=U$jKq
% [theta,r] = cart2pol(X,Y); S92!jp/
% idx = r<=1; 6u]OXPA|
% z = nan(size(X)); UdM5R
[
% n = [0 1 1 2 2 2 3 3 3 3]; [7Kj$PB3
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (/rIodHJO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ~)\1g0
% y = zernfun(n,m,r(idx),theta(idx)); -^nQ^Td=j
% figure('Units','normalized') :O@,Z_"
% for k = 1:10 Q/9vDv
% z(idx) = y(:,k); ]6c2[r?g{
% subplot(4,7,Nplot(k)) l8n[8AT1
% pcolor(x,x,z), shading interp TQxc?o
% set(gca,'XTick',[],'YTick',[]) 5F_:[H =
% axis square ^Ihdq89 t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) B#V4
% end V44sNi
% hcqmjqJ
% See also ZERNPOL, ZERNFUN2. `a1R "A
Dm`U|<o
% Paul Fricker 11/13/2006 _$jJpy
HI`A;G]
9QM"JEu@
% Check and prepare the inputs: 0R!}}*Ee>q
% ----------------------------- $R#L@iL-
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .^+$w$
error('zernfun:NMvectors','N and M must be vectors.') fm2M i~}0
end u C8T!z
_/w-gL{
if length(n)~=length(m) +H^V},dBp!
error('zernfun:NMlength','N and M must be the same length.') X72X:"
end OQb9ijLeK
fyoB]{$p8
n = n(:); ^DCv-R+p
m = m(:); co%_~xO
if any(mod(n-m,2)) 9p'J(`
error('zernfun:NMmultiplesof2', ... Dp |FyP_w
'All N and M must differ by multiples of 2 (including 0).') o%JIJ7M
end V$F.`O!hfi
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if any(m>n) FoXQ]X7"
error('zernfun:MlessthanN', ... EF^=3
'Each M must be less than or equal to its corresponding N.') 0*M}QXt
end umn~hb5O
qO3BQ]UF
if any( r>1 | r<0 ) 1kw4'#J8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') U$JIF/MO_
end ^{+:w:g
*t*&Q /W
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y/I6.K3
error('zernfun:RTHvector','R and THETA must be vectors.') DT]p14@t9
end |Ie`L("
m-FDCiN>
r = r(:); 2}C>{*}yQ
theta = theta(:); ->9xw
length_r = length(r); 1Moh`
if length_r~=length(theta) *xVAm7_v
error('zernfun:RTHlength', ... x{o5Ha{
'The number of R- and THETA-values must be equal.') SpiC0
end cZT.vA#
/<(ik&%N
% Check normalization: U jzz`!mz
% -------------------- 3NZFW{u
if nargin==5 && ischar(nflag) xVX||rrh
isnorm = strcmpi(nflag,'norm'); Yf`.Cq_:
if ~isnorm '*Mb
.s"
error('zernfun:normalization','Unrecognized normalization flag.') &+iW:
end R*fR?
else % x;!s=U
isnorm = false; Hu2g (!
end 'yjH~F.
trt\PP:H%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n
k3lC/f
% Compute the Zernike Polynomials &nw~gSe
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /\I%)B47^9
''07Km@x
% Determine the required powers of r: ;7*@Gf}R
% ----------------------------------- 0!
%}
m_abs = abs(m); shvcc
rpowers = []; <&Xq`i/(
for j = 1:length(n) 7V``f:#d
rpowers = [rpowers m_abs(j):2:n(j)]; %"fKZ
end m6<0 hP
rpowers = unique(rpowers); Q8:ocEhR
; O0rt1
% Pre-compute the values of r raised to the required powers, ,X6j$YLWp
% and compile them in a matrix: dph6aN(49
% ----------------------------- agD.J)v\
if rpowers(1)==0 `I{Q,HQ7
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); CxQ,yd;>
rpowern = cat(2,rpowern{:}); ha~s<
I
rpowern = [ones(length_r,1) rpowern]; n9-[z2n
else N\&;R$[9:
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MoHvXp;X
rpowern = cat(2,rpowern{:}); |:[vpJFK
end a[ l5k
R?SHXJ%'
% Compute the values of the polynomials: .w)t<7 y
% -------------------------------------- ^`?>
Huu<w
y = zeros(length_r,length(n)); +[`%b3N k
for j = 1:length(n) 0E1)&f
s = 0:(n(j)-m_abs(j))/2; }`FPe
pows = n(j):-2:m_abs(j); _S1uJ~j;E
for k = length(s):-1:1 k<qH<<r*
p = (1-2*mod(s(k),2))* ... zSCPp6
prod(2:(n(j)-s(k)))/ ... <2d@\"AoHE
prod(2:s(k))/ ... e84TLU?~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... hDsORh!i
prod(2:((n(j)+m_abs(j))/2-s(k))); 2jC\yY |PN
idx = (pows(k)==rpowers); ]Jqe)o
y(:,j) = y(:,j) + p*rpowern(:,idx); Z.JTq~`I
end l si8?91
.#|pje^
if isnorm k#[s)Ja?s
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4/|=0TC;
end KW<CU'
end _R6> Ayw*
% END: Compute the Zernike Polynomials I),8EEf\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZeZwzH)BD
Wz]S+IpY
% Compute the Zernike functions: .5xM7,
% ------------------------------ ]"6<"1)
idx_pos = m>0; bHnQLJ
idx_neg = m<0; IIZsN*^
l!,{bOZ
z = y; 2Oa-c|F
if any(idx_pos) B"v=Fr[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M-gjS6c\3
end 9n'p 7(s%
if any(idx_neg) +n dyR
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %Z4=3?5B"9
end < r~Tj
!Ao?bs'
% EOF zernfun