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 ^DS9D:oE
function z = zernfun(n,m,r,theta,nflag) 6k%N\!_TUW
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;E l"dqH
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #[vmS
% and angular frequency M, evaluated at positions (R,THETA) on the 4xk'R[v
% unit circle. N is a vector of positive integers (including 0), and 36,qh.LKn
% M is a vector with the same number of elements as N. Each element Qf6]qJa|
% k of M must be a positive integer, with possible values M(k) = -N(k) INby0S
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, CN#`m]l.
% and THETA is a vector of angles. R and THETA must have the same K
4j'e6
% length. The output Z is a matrix with one column for every (N,M) :O-Y67>&
% pair, and one row for every (R,THETA) pair. 3v
:PBmE
% HDvj{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike SouPk/-B80
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )}\jbh>RH
% with delta(m,0) the Kronecker delta, is chosen so that the integral G#ZU^%$M,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3+u11'0=t
% and theta=0 to theta=2*pi) is unity. For the non-normalized - U!:.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ajq [ID
% cF_ Y}C
% The Zernike functions are an orthogonal basis on the unit circle. rBye%rQRq
% They are used in disciplines such as astronomy, optics, and (/1 4)"Sk
% optometry to describe functions on a circular domain. |lm
% P#\L6EO.
% The following table lists the first 15 Zernike functions. |Kky+*
% +v2Fr}
% n m Zernike function Normalization +e);lS"+/
% -------------------------------------------------- Q@6OIE
% 0 0 1 1 v
T2YX5k&,
% 1 1 r * cos(theta) 2 !e*Q2H+
% 1 -1 r * sin(theta) 2 Bf~
% 2 -2 r^2 * cos(2*theta) sqrt(6) `YVdIDl]
% 2 0 (2*r^2 - 1) sqrt(3) ;Xk-hhR
% 2 2 r^2 * sin(2*theta) sqrt(6) L(XGD
% 3 -3 r^3 * cos(3*theta) sqrt(8) 0(VAmb%{
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) hn{]Q@(I
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) xbnx*4o0
% 3 3 r^3 * sin(3*theta) sqrt(8) ~J^Gzl
% 4 -4 r^4 * cos(4*theta) sqrt(10) 1q0DOf]!T
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A6v02WG_1T
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) }]$%aMxy T
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -"YQo
% 4 4 r^4 * sin(4*theta) sqrt(10) `of 5h*k
% -------------------------------------------------- \`}Rdr!p%
% W(Z_ac^e[
% Example 1: 7dyGC:YuTL
% i
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% % Display the Zernike function Z(n=5,m=1) mRZC98$ @r
% x = -1:0.01:1; X|^E+
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% [X,Y] = meshgrid(x,x); 7(rNJPrU~=
% [theta,r] = cart2pol(X,Y); tsVQXvo
% idx = r<=1; _)
k=F=
% z = nan(size(X)); 0ubT/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); mnZ/rb
% figure td%]l1
% pcolor(x,x,z), shading interp ()e.J
% axis square, colorbar O]>9\!0{
% title('Zernike function Z_5^1(r,\theta)') :0|]cHm
% Tqz{{]%j~$
% Example 2: S1sNVW
% 3}e-qFlV8,
% % Display the first 10 Zernike functions #_0OYL`(mE
% x = -1:0.01:1; nd*9vxM
% [X,Y] = meshgrid(x,x); {G&*\5W
% [theta,r] = cart2pol(X,Y); `WQz_}TqB
% idx = r<=1; {XH!`\
% z = nan(size(X)); 1wP#?p)c
% n = [0 1 1 2 2 2 3 3 3 3]; =cI -<0QSn
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ,r~pf(nz
% Nplot = [4 10 12 16 18 20 22 24 26 28]; GjN/8>/
% y = zernfun(n,m,r(idx),theta(idx)); *yKw@@d+p
% figure('Units','normalized') &
9}L +/,
% for k = 1:10 4scY8(1
% z(idx) = y(:,k); G8dC5+h
% subplot(4,7,Nplot(k)) Sm(X/P=z
% pcolor(x,x,z), shading interp EvSo|}JA[
% set(gca,'XTick',[],'YTick',[]) c]LE9<G
% axis square R#gt~]x6k
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RnC96"";R.
% end cK4Q! l6O
% 0NrUB
% See also ZERNPOL, ZERNFUN2. 'X_8j` ]#
is}6cR
% Paul Fricker 11/13/2006 `>KB8SY:qK
PDQC^2Z
3Kuu9<0
% Check and prepare the inputs: CeQL8yJ;
% ----------------------------- Ks'msSMC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) GcN[bH(@
error('zernfun:NMvectors','N and M must be vectors.') J&Ig%&/
end 0?OTa<c
)7!q>^S{B
if length(n)~=length(m) lZ]x #v
error('zernfun:NMlength','N and M must be the same length.') NwPGH=V
end 5-'jYp/
:U;n?Zu
S
n = n(:); `/?XvF\
m = m(:); _`3'D`s
if any(mod(n-m,2)) sjl(
error('zernfun:NMmultiplesof2', ... mU0j K@^&M
'All N and M must differ by multiples of 2 (including 0).') &/QdG= r +
end XgRrJ.
tgmG#b*
if any(m>n) \yt-_W=[
error('zernfun:MlessthanN', ... L3}n(KAJj
'Each M must be less than or equal to its corresponding N.') /g$cQ=c
end 3ht>eaHi
qJV2x.!
if any( r>1 | r<0 ) yKupPp);
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,@I_b
end {l/j?1Dxq
-M=#U\D
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) C>l{_J)n
error('zernfun:RTHvector','R and THETA must be vectors.') |,!]]YO.V
end R>Q&Ax
|e{F;8
r = r(:); {2jetX`@h
theta = theta(:); !J#oN+AR
length_r = length(r); 9vIqGz-o
if length_r~=length(theta) }U <T>0
error('zernfun:RTHlength', ... #?=?<"*j
'The number of R- and THETA-values must be equal.') ((KNOa5
end Y2lBQp8'|
2cv!85
% Check normalization: X}"Ic@8
% -------------------- aC$-riP,?'
if nargin==5 && ischar(nflag) Tfasry9'8
isnorm = strcmpi(nflag,'norm'); %LI[+#QE
if ~isnorm 2AYV9egZ
error('zernfun:normalization','Unrecognized normalization flag.') 9Q\CJ9
end 3PRg/vD3
else o8<0#W@S
isnorm = false; q{4W@Um-
end t<8vgdD
RWyDX_z#<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZiR },F/
% Compute the Zernike Polynomials RP!!6A6:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Js2/s
8&[Lr o9
% Determine the required powers of r: d yH<D5
% ----------------------------------- 9, A(|g
m_abs = abs(m); 7Iz%Jty
rpowers = []; ;4(ULJ*
for j = 1:length(n) Kjw==5)}
rpowers = [rpowers m_abs(j):2:n(j)]; n8h1SlK08
end +#* F"k(
rpowers = unique(rpowers); r'|V z*/h
kmNa),`{s
% Pre-compute the values of r raised to the required powers, [p& n]T
% and compile them in a matrix: sR~D3-
% ----------------------------- ]o!rK<
if rpowers(1)==0 :? uUh
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); s&Bk@a8
rpowern = cat(2,rpowern{:}); ,)&ansN
rpowern = [ones(length_r,1) rpowern];
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else IKz3IR eu
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )Cas0~ RM
rpowern = cat(2,rpowern{:}); f$7Xh~
end ""~b1kEt
2OA0rH"v
% Compute the values of the polynomials: z (1zth
% -------------------------------------- qGlbO
y = zeros(length_r,length(n)); Fx@ovI- 5
for j = 1:length(n) !xE/
s = 0:(n(j)-m_abs(j))/2; ]n \Qa
pows = n(j):-2:m_abs(j); Xu.Wdl/{Ra
for k = length(s):-1:1 LqYP0%7
p = (1-2*mod(s(k),2))* ... c[IT?6J4
prod(2:(n(j)-s(k)))/ ... dnwTD\),
prod(2:s(k))/ ... Ym% $!#
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 96(3ilAt
prod(2:((n(j)+m_abs(j))/2-s(k))); pA%}CmrMq
idx = (pows(k)==rpowers); TTDcVG_}
y(:,j) = y(:,j) + p*rpowern(:,idx); Pv#Oea?
end l1 M
%
I ~U1vtgp
if isnorm R^p'gQc$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); k^H&IS!
end B|f
=hlY
end 3-=f@uH!
% END: Compute the Zernike Polynomials c 5%uiv]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (yJY/|
N1',`L5
% Compute the Zernike functions: =8o$
% ------------------------------ ^@V;`jsll
idx_pos = m>0; "^froQ{"T
idx_neg = m<0; \ 4`:~c
)X2/_3
z = y; =K\xE"
if any(idx_pos)
DXa!"ZU
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k#g` n3L
end {py"Ob_
if any(idx_neg) g7UZtpLTm
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &E?TR
A# E
end & FpoMW
>iV2>o _
% EOF zernfun