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 s''?:
+
function z = zernfun(n,m,r,theta,nflag) :=vB|Ch:~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. P.Pw.[:3
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <6^MVaD
% and angular frequency M, evaluated at positions (R,THETA) on the y%)5r}S^
% unit circle. N is a vector of positive integers (including 0), and EM]~yn!+
% M is a vector with the same number of elements as N. Each element ?#?[6t
% k of M must be a positive integer, with possible values M(k) = -N(k) Dz/I"bZLC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Sp$~)f'
% and THETA is a vector of angles. R and THETA must have the same Z*S
9pkWcF
% length. The output Z is a matrix with one column for every (N,M) | n5F_RL
% pair, and one row for every (R,THETA) pair. m<)0XE6w
% l<5O\?Vo]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike N|hNh$J[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), v(D{_
% with delta(m,0) the Kronecker delta, is chosen so that the integral Qb}7lm{r
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, OrP-+eg
% and theta=0 to theta=2*pi) is unity. For the non-normalized n^P=a'+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BE. v+'c"
% )R$+dPu>
% The Zernike functions are an orthogonal basis on the unit circle. 9z7^0Ruw
% They are used in disciplines such as astronomy, optics, and 2`>/y
% optometry to describe functions on a circular domain. 7NC"}JB&
% }@MOkj
% The following table lists the first 15 Zernike functions. U ^1Xc#Ff
% p'UY Ht
% n m Zernike function Normalization & V:q}Q
% -------------------------------------------------- tu\;I{h=0
% 0 0 1 1 D>M
a3g
% 1 1 r * cos(theta) 2 #-
$?2?2
% 1 -1 r * sin(theta) 2 )1ia;6}
% 2 -2 r^2 * cos(2*theta) sqrt(6) <,p|3p3
% 2 0 (2*r^2 - 1) sqrt(3) L4`bGZl55
% 2 2 r^2 * sin(2*theta) sqrt(6) Qr]xj7\@i
% 3 -3 r^3 * cos(3*theta) sqrt(8) _[;>V*?zp5
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) N:'v^0
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) `:cnu;
% 3 3 r^3 * sin(3*theta) sqrt(8) p\I,P2on
% 4 -4 r^4 * cos(4*theta) sqrt(10) #mg6F$E
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x*td
nor&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tdSy&]P
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %,f|H :+>u
% 4 4 r^4 * sin(4*theta) sqrt(10) KrE:ilm#^Y
% -------------------------------------------------- )W9W8>Cc5_
% i? 5jl&30
% Example 1: taOD,}c|$
% [of{~
% % Display the Zernike function Z(n=5,m=1) `|K30hRp:
% x = -1:0.01:1; O{Bll;C
% [X,Y] = meshgrid(x,x); 5W"&$6vj
% [theta,r] = cart2pol(X,Y); K6<@DP+/
% idx = r<=1; E!<w t
% z = nan(size(X)); ,l`4)@{G
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _j{^I^P
% figure sv`+?hjG
% pcolor(x,x,z), shading interp .;j} :<
% axis square, colorbar rFJ(t7\9h
% title('Zernike function Z_5^1(r,\theta)') QX}O{LQR
% %^){Z,}M}
% Example 2: gi!{y
% !G E-5 \*
% % Display the first 10 Zernike functions X,VOKj.%
% x = -1:0.01:1; =4`#OQ&g
% [X,Y] = meshgrid(x,x); |uo<<-\jTO
% [theta,r] = cart2pol(X,Y); P 1`X<A
% idx = r<=1; gN#&Ag<?
% z = nan(size(X)); XnC`JO+7M
% n = [0 1 1 2 2 2 3 3 3 3]; \49LgN@\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; vi?{H*H4c
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ! bbVa/
% y = zernfun(n,m,r(idx),theta(idx)); o,l 3j|1
% figure('Units','normalized') u,AZMjlF
% for k = 1:10 [1{#a {4
% z(idx) = y(:,k); ZL[~[
% subplot(4,7,Nplot(k)) 9x;CJhX
% pcolor(x,x,z), shading interp ^q``f%Xt
% set(gca,'XTick',[],'YTick',[]) 0<f\bY02
% axis square R'RLF
=
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) qOaI4JP@
% end RC(fhqV
% 57r?`'#*
% See also ZERNPOL, ZERNFUN2. r #H(kJu,
^M?O
% Paul Fricker 11/13/2006 !ceT>i90h
LASR*
cHN
eiOF
% Check and prepare the inputs: E}eu]2=nU}
% ----------------------------- q{D_p[q
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7_t\wmvYp
error('zernfun:NMvectors','N and M must be vectors.') lq0@)'D
end S[!sJ-rG
AQX~do\A
if length(n)~=length(m) ^6&?R?y
error('zernfun:NMlength','N and M must be the same length.') -*q:B[d
end bvHF;Qywg
'iy &%?
n = n(:); ",wv*z)_>
m = m(:); paFiuQ
if any(mod(n-m,2)) xWkCP2$?P
error('zernfun:NMmultiplesof2', ... :4 9ttJl
'All N and M must differ by multiples of 2 (including 0).') #H9J/k_
end 'N1_:$z@(
4`Com~`6"
if any(m>n) aju!A q54G
error('zernfun:MlessthanN', ... JP$@*F@t
'Each M must be less than or equal to its corresponding N.') {2u#Q7]|
end 6J/"1_
aD
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if any( r>1 | r<0 ) Ejc%DSG
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7R9S%
end fq*.4s
#
|#<PI9)`
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /8V#6d_
error('zernfun:RTHvector','R and THETA must be vectors.') ^kg[n908Nw
end y$6m|5
)$18a
r = r(:); AhjK*nJF
theta = theta(:); );4lM%]eb
length_r = length(r); 8?ig/HSt2
if length_r~=length(theta) vZEeb j
error('zernfun:RTHlength', ... tqI]S
X
'The number of R- and THETA-values must be equal.') X\X*-.]{
end gFk~SJd
q5X\wz2N
% Check normalization: py9zDWk~
% -------------------- (r8Rb*OP
if nargin==5 && ischar(nflag) J;Y=oB
isnorm = strcmpi(nflag,'norm'); 5nq0#0Oc
if ~isnorm hh\\api
error('zernfun:normalization','Unrecognized normalization flag.') H>8B$fi )$
end =,Yi" E
else +T}:GBwD7
isnorm = false; L2"fO
end !>$tRW?gH~
|7@[+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *=]hc@
% Compute the Zernike Polynomials pJM~'tlHV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p-]vf$u
3+6s}u)
% Determine the required powers of r:
D{hsa
% ----------------------------------- 9 *>@s
m_abs = abs(m); ~*-(_<FH
rpowers = []; L_f u<W
for j = 1:length(n) @ 2r9JqR[=
rpowers = [rpowers m_abs(j):2:n(j)]; X+l&MD
end /S29\^
rpowers = unique(rpowers); <~9z.v7
H{*~d+:ol
% Pre-compute the values of r raised to the required powers, bLMN9wGOgK
% and compile them in a matrix: BE
n$~4-
% ----------------------------- j^DoILw
if rpowers(1)==0 0fgt2gA33
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $N$
ZJC6(@
rpowern = cat(2,rpowern{:}); .h)o\6Wq
rpowern = [ones(length_r,1) rpowern]; Lf+M
+^l
else }UwDHq=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2z/qbzG7
rpowern = cat(2,rpowern{:}); dZnAdlJ
end xf1@mi[a
x1['+!01
% Compute the values of the polynomials: 9|yn{4E
% -------------------------------------- GX4HW \>a
y = zeros(length_r,length(n)); \!HGkmd
for j = 1:length(n) S{cy|QD
s = 0:(n(j)-m_abs(j))/2; 6?-vj2,
pows = n(j):-2:m_abs(j); ?yKW^,q+
for k = length(s):-1:1 w_-v!s2
p = (1-2*mod(s(k),2))* ... 5mNd5IM
prod(2:(n(j)-s(k)))/ ... CRy;>UI
prod(2:s(k))/ ... (rfU=E
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H]@M00C
prod(2:((n(j)+m_abs(j))/2-s(k))); /A3tY"Vn
idx = (pows(k)==rpowers); jkd'2
y(:,j) = y(:,j) + p*rpowern(:,idx); +bwSu)k
end Hm=!;xAFX
0pP;[7k\
if isnorm BElVkb
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #DMt<1#:
end HorFQ?8
end =,B44:`r
% END: Compute the Zernike Polynomials T;(k
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
M1><K:
BbA7X
% Compute the Zernike functions: h
WvQh
% ------------------------------ Obd@#uab
idx_pos = m>0; # biI=S
idx_neg = m<0; c]]OV7;)>
hS)X`M
z = y; Z1j3 F
if any(idx_pos) 9pN},F91n:
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]qZs^kQ
end __Kn 1H{
if any(idx_neg) BM+v,hGY
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); O)g\/uRy
end .Y}~2n
m
Cvgs
% EOF zernfun