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 n[!;yO
function z = zernfun(n,m,r,theta,nflag) 6cM<>&e
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. y|$R`P
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0,HqE='w
% and angular frequency M, evaluated at positions (R,THETA) on the 7ftR4
% unit circle. N is a vector of positive integers (including 0), and \gLxC
% M is a vector with the same number of elements as N. Each element qAoAUDm
% k of M must be a positive integer, with possible values M(k) = -N(k) i+g~ Uj}h
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =I2@/,
% and THETA is a vector of angles. R and THETA must have the same #~L!pKM
% length. The output Z is a matrix with one column for every (N,M) R (G2qi
% pair, and one row for every (R,THETA) pair. |,b2b2v?
% z~,mRgc$B
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $9K(F~/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U4BqO
:sd
% with delta(m,0) the Kronecker delta, is chosen so that the integral Yu'a<5f
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4'' ,6KJ@
% and theta=0 to theta=2*pi) is unity. For the non-normalized e}c&LDgU
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. dL-i)F
% NUCiY\td
% The Zernike functions are an orthogonal basis on the unit circle. ._?V%/
% They are used in disciplines such as astronomy, optics, and zh\$t]d<I
% optometry to describe functions on a circular domain. @5xu>g Kn
% Z7fg
25
% The following table lists the first 15 Zernike functions. sYJL-2JX
% .u l
53 m
% n m Zernike function Normalization yub{8 f;v
% -------------------------------------------------- mzWP8Hlw
% 0 0 1 1 }Dn^d}?s||
% 1 1 r * cos(theta) 2 CK0l9#g
% 1 -1 r * sin(theta) 2 Us,)]W.S
% 2 -2 r^2 * cos(2*theta) sqrt(6) `\bT'~P
% 2 0 (2*r^2 - 1) sqrt(3) \q "N/$5{f
% 2 2 r^2 * sin(2*theta) sqrt(6) RT^v:paNT2
% 3 -3 r^3 * cos(3*theta) sqrt(8) `5q
;ssu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {T=52h=e
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) OR:[J5M)
% 3 3 r^3 * sin(3*theta) sqrt(8) v?%LQKO
% 4 -4 r^4 * cos(4*theta) sqrt(10) 3GF2eS$$P
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /`[!_4i
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _%~$'Hy
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D8%AV;-Y
% 4 4 r^4 * sin(4*theta) sqrt(10) 03k?:D+5
% -------------------------------------------------- "X04mQn15
% WNs}sNSf
% Example 1: i^)WPP>4Aw
% K B!5u 9
% % Display the Zernike function Z(n=5,m=1) YuQ~AE'i
% x = -1:0.01:1; 6.5wZN9<|
% [X,Y] = meshgrid(x,x); +f>c xA
% [theta,r] = cart2pol(X,Y); & ze>X
% idx = r<=1; TW7:q83{l
% z = nan(size(X)); d,0 }VaY=D
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Zp&@h-%YoD
% figure (gwj)?:
% pcolor(x,x,z), shading interp s
=Umj'1k
% axis square, colorbar eS'yGY0b
% title('Zernike function Z_5^1(r,\theta)') vi!YN|}\
% S{#cD1>.
% Example 2: FY'ty@|_s
% u,1}h L
% % Display the first 10 Zernike functions j}:~5 |.
% x = -1:0.01:1; x[Im%k
% [X,Y] = meshgrid(x,x); k`\R+WK$
% [theta,r] = cart2pol(X,Y); >\2:\wI
% idx = r<=1; [8XLK 4e
% z = nan(size(X)); 8z2Rry
w
% n = [0 1 1 2 2 2 3 3 3 3]; ?+0GfIV
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; e5?PkFV^a1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; n6MM5h/#r
% y = zernfun(n,m,r(idx),theta(idx)); uuNR?1fS
% figure('Units','normalized') WC,+Cn e
% for k = 1:10 ?F7o!B
% z(idx) = y(:,k); rJJ[X4$
% subplot(4,7,Nplot(k)) MFt*&%,JX
% pcolor(x,x,z), shading interp .]x2K-Sf
% set(gca,'XTick',[],'YTick',[]) -|S]oJy
% axis square LD>\#q8a*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;eL9{eF
% end *\uM.m0$
% ememce,Np
% See also ZERNPOL, ZERNFUN2. =1p8i
8RW&r
% Paul Fricker 11/13/2006 Q`%R[#
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% Check and prepare the inputs: crcA\lJf
% ----------------------------- tV;`fV
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6w[}&pX"z
error('zernfun:NMvectors','N and M must be vectors.') q.#[TI ^
end S6d`ioi-
\x{;U#B[3>
if length(n)~=length(m) )B#
,
error('zernfun:NMlength','N and M must be the same length.') errH>D~
end Pmg)v!"
sP@X g;]
n = n(:); .|qK+Hnc
m = m(:); mmXm\]r>4
if any(mod(n-m,2)) v``-F(i$
error('zernfun:NMmultiplesof2', ... U69u'G:
'All N and M must differ by multiples of 2 (including 0).') ;Q;[*B=kE
end -]uUY e
c
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if any(m>n) +!IQj0&'Y3
error('zernfun:MlessthanN', ... ~[WF_NU1y
'Each M must be less than or equal to its corresponding N.') gi/@j
end )d\j I
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if any( r>1 | r<0 ) A>`945|
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ?~"bR%
end g>rp@M
YTQt3=1ii
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) }9HmTr|
error('zernfun:RTHvector','R and THETA must be vectors.') kum#^^4G|
end 'ly?P8h
vbx6I>\Y
r = r(:); 7?8wyk|x
theta = theta(:); 9^"b*&>P
length_r = length(r); #`TgZKDg2
if length_r~=length(theta) =<r8fXWZ
error('zernfun:RTHlength', ... mR\`DltoV
'The number of R- and THETA-values must be equal.') {Gq*e/
end kE8>dmH23
s>k Uh
% Check normalization: &6 s) X
% -------------------- ml0.$z
if nargin==5 && ischar(nflag) QxuhGA
isnorm = strcmpi(nflag,'norm'); }8|[;Qa`y
if ~isnorm E!BPE>
error('zernfun:normalization','Unrecognized normalization flag.') ]M/9#mD9~
end pLa[}=
else Z=B_Ty
isnorm = false; E:zF/$tG
end %*aJLn+]_R
b*a2,MiM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S##1GOO
% Compute the Zernike Polynomials :@W.K5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *<N3_tx"
6qN~/TnHZ
% Determine the required powers of r: 6u`F
d#
% ----------------------------------- 2%*MW"Q
m_abs = abs(m); )"zvwgaW
rpowers = []; <FMq>d$\
for j = 1:length(n) c_aZ{S
rpowers = [rpowers m_abs(j):2:n(j)]; iGB_{F~t4}
end Uv
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rpowers = unique(rpowers); ~\x:<)
RLlU"
sw+{
% Pre-compute the values of r raised to the required powers, O }9KJU
% and compile them in a matrix: (b?{xf'G
% ----------------------------- X[#zCM
if rpowers(1)==0 *
tCS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 08X_}97#WF
rpowern = cat(2,rpowern{:}); Pe C7
rpowern = [ones(length_r,1) rpowern]; !O\;Nua
else [E#UGJ@
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [."[pY
rpowern = cat(2,rpowern{:}); 8WE{5#oi
end %Qg+R26U
5es[Ph|K5
% Compute the values of the polynomials: J=.`wZQkS
% -------------------------------------- Rqwzh@}
y = zeros(length_r,length(n)); UAR5^
for j = 1:length(n) ^[%%r3"$C
s = 0:(n(j)-m_abs(j))/2; eC5 $#,HiC
pows = n(j):-2:m_abs(j); 6wco&7
for k = length(s):-1:1 zF5uN:-s
p = (1-2*mod(s(k),2))* ... $/6;9d^
prod(2:(n(j)-s(k)))/ ... QwhRNnE=
prod(2:s(k))/ ... l5l>d62
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w9
w%&{j
prod(2:((n(j)+m_abs(j))/2-s(k))); e><5Pr)
idx = (pows(k)==rpowers); G=;k=oX(
y(:,j) = y(:,j) + p*rpowern(:,idx); >~`C-K#
end Kwc6mlw~M
s2j['g5
if isnorm .]aF
1}AI
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); x0d~i!d
end Bgmn2-
end Ra*e5
% END: Compute the Zernike Polynomials }j,[ 1@S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JCAq8=zM
JG{j)O|L
% Compute the Zernike functions: L
8{\r$
% ------------------------------ eY{+~|KZ
idx_pos = m>0; 7JSNYTH
idx_neg = m<0; .9O$G2'oh
EUsI%p
z = y; D&HV6#
if any(idx_pos) (E]!Z vE
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]Qm]I1P
end 0Z{j>=$
if any(idx_neg) czlFr|O;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); eT2*W$
end s+:=I
e
5>AX*]c
% EOF zernfun