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 A/ 88WC$v
function z = zernfun(n,m,r,theta,nflag) 7,5Bur
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z^_gS&nDa~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N YU/?AQg
% and angular frequency M, evaluated at positions (R,THETA) on the F $1f8U8
% unit circle. N is a vector of positive integers (including 0), and 1EA#c>I$
% M is a vector with the same number of elements as N. Each element p;.M.
% k of M must be a positive integer, with possible values M(k) = -N(k) 5Tq*]ZE
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :r9<wbr)k0
% and THETA is a vector of angles. R and THETA must have the same b!`{fwV
% length. The output Z is a matrix with one column for every (N,M) zQaD&2 q
% pair, and one row for every (R,THETA) pair. l;}3J3/qq]
% hd@jm^k
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $) m$c5!
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -mLS\TF S
% with delta(m,0) the Kronecker delta, is chosen so that the integral LpN3cy>U
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ix+eP|8F
% and theta=0 to theta=2*pi) is unity. For the non-normalized vF1Fcp.@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. x.Tulo0/
% -lm)xpp1
% The Zernike functions are an orthogonal basis on the unit circle. Lwn
% They are used in disciplines such as astronomy, optics, and (h'Bz6K
% optometry to describe functions on a circular domain. pKaU
[1x?%
% 'PWA
% The following table lists the first 15 Zernike functions. S
^$!n,
% DGNn#DP
% n m Zernike function Normalization __}ut+H^5p
% -------------------------------------------------- {%c&T S@s
% 0 0 1 1 b*1yvkX5
% 1 1 r * cos(theta) 2 2WC$r8E
% 1 -1 r * sin(theta) 2 ]EdZ,`B4
% 2 -2 r^2 * cos(2*theta) sqrt(6) B[9y<FB+
% 2 0 (2*r^2 - 1) sqrt(3) fNz(z\
% 2 2 r^2 * sin(2*theta) sqrt(6) wlgR =l
% 3 -3 r^3 * cos(3*theta) sqrt(8) UjS+Ddp
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) r+;k(HMY}[
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) OAf}\
% 3 3 r^3 * sin(3*theta) sqrt(8) Yz#E0aTTA
% 4 -4 r^4 * cos(4*theta) sqrt(10) d'iSvd.
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /}9)ZYMx
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) X.ecA`0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }m&\I
% 4 4 r^4 * sin(4*theta) sqrt(10) Po*!eD
% -------------------------------------------------- 3n~O&{
% rE]Nr ;Ys
% Example 1: E;wT4 T=
% i|m8#*Hd
% % Display the Zernike function Z(n=5,m=1) #x`K4f)
% x = -1:0.01:1; ae" o|Q
% [X,Y] = meshgrid(x,x); "z*.Bk
% [theta,r] = cart2pol(X,Y); sDAP'&
% idx = r<=1; sf/m@425
% z = nan(size(X)); ESUO I
% z(idx) = zernfun(5,1,r(idx),theta(idx)); =cO5Nt
% figure [5tvdW6Z&
% pcolor(x,x,z), shading interp %J Jp/I
% axis square, colorbar R+z'6&/ =I
% title('Zernike function Z_5^1(r,\theta)') LH.Gf
% >'4$g7o,
% Example 2: W =;,ls
% Y=?{TX=6<[
% % Display the first 10 Zernike functions 4>OS2b`.;
% x = -1:0.01:1; K1o>>388G
% [X,Y] = meshgrid(x,x); Xu
E' %;:
% [theta,r] = cart2pol(X,Y); C#e :_e]
% idx = r<=1; +~
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% z = nan(size(X));
;u[:J
% n = [0 1 1 2 2 2 3 3 3 3]; q%QvBN
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Hzj8o3
% Nplot = [4 10 12 16 18 20 22 24 26 28]; %?, 7!|Ls
% y = zernfun(n,m,r(idx),theta(idx)); ^$}O?y7O
% figure('Units','normalized') }V*?~.R
% for k = 1:10 ` &bF@$((
% z(idx) = y(:,k); V)`A,7X
% subplot(4,7,Nplot(k)) j}d):3!
% pcolor(x,x,z), shading interp _|W&tB*
% set(gca,'XTick',[],'YTick',[]) [PB73q8
% axis square Pksr9"Ah
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) e$_gOwB
% end B8?9L8M}
% kZo#Ny
% See also ZERNPOL, ZERNFUN2. H }]Zp
""
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% Paul Fricker 11/13/2006 Q)BSngW+
]kx<aQ^
S.Kcb=;"L
% Check and prepare the inputs: 0Ze&GK'Hf
% ----------------------------- ,YjjL
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vea{o35!
error('zernfun:NMvectors','N and M must be vectors.') _
Pzgn@D
end 8+dsTX`|S
P^=B6>e
if length(n)~=length(m) ,/GFD[SQ
error('zernfun:NMlength','N and M must be the same length.') SmD#hE[
end jtpHDS
)m3emMO2
n = n(:); PX_9i@ZG
m = m(:); Og1\6Q
if any(mod(n-m,2)) &Ep$<kx8
error('zernfun:NMmultiplesof2', ... ":
BZZ\!
'All N and M must differ by multiples of 2 (including 0).') R[6R)#o
end \Gk}Fer
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if any(m>n) 0PFC%x
error('zernfun:MlessthanN', ... B8T5?bl
'Each M must be less than or equal to its corresponding N.') yx&}bu\
end ^`dMjeF
.pe.K3G&
if any( r>1 | r<0 ) m(:R (K(je
error('zernfun:Rlessthan1','All R must be between 0 and 1.') eYoc(bG(+
end ZVJ6 {DS/
CdCY#$Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (zy|>u
error('zernfun:RTHvector','R and THETA must be vectors.') g#l!b%$
end I]5){Q"S
}j1;0 kb?
r = r(:); CE
theta = theta(:); Gdx%#@/
length_r = length(r); :.l\lj0Yf
if length_r~=length(theta) C"`\[F`.k
error('zernfun:RTHlength', ... QD^= ;!
'The number of R- and THETA-values must be equal.') 5>CeFy
end RT'5i$q[
v,N!cp1
% Check normalization: kO^
% -------------------- i@WO>+iB
if nargin==5 && ischar(nflag) !@Vj&>mH$
isnorm = strcmpi(nflag,'norm'); ak3WER|f#
if ~isnorm qkc,93B3
error('zernfun:normalization','Unrecognized normalization flag.') S\sy^Kt~4:
end &1=,?s]&
else Bqa_l|
isnorm = false; K)`R?CZ:s
end .3Smqwm=Y
:mCGY9d4L
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wod{C !
% Compute the Zernike Polynomials { i3x\|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *"F*6+}w"
hZUS#75M5
% Determine the required powers of r: TQ/#
% ----------------------------------- X,o ]tgg=
m_abs = abs(m); GO][`zZJ]
rpowers = []; jamai8
for j = 1:length(n) #&S<{75A
rpowers = [rpowers m_abs(j):2:n(j)]; JPT&!%~
end ]>sMu]biH
rpowers = unique(rpowers); .1J`>T?=Q
1ATH$x
% Pre-compute the values of r raised to the required powers, e*Nm[*@UW
% and compile them in a matrix: p{r{}iYI
% ----------------------------- HQ4WunH2Y
if rpowers(1)==0 c[OQo~m$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); k^k1>F}yx
rpowern = cat(2,rpowern{:}); T_)+l)
rpowern = [ones(length_r,1) rpowern]; cY~lDLyB
else )0;O<G] d
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); c No)LF
rpowern = cat(2,rpowern{:}); <bv9X?U
end l~kxK.Ru
b_~KtMO
% Compute the values of the polynomials: ={ 190=\9
% -------------------------------------- MD> E0p)
y = zeros(length_r,length(n)); rHjR 4q
for j = 1:length(n) !a5e{QG0
s = 0:(n(j)-m_abs(j))/2; #]} G{
P
pows = n(j):-2:m_abs(j); =`gFwH<
for k = length(s):-1:1 1EV0Y]T1
p = (1-2*mod(s(k),2))* ... Uf[Gs/!NV
prod(2:(n(j)-s(k)))/ ... /h&>tYVio
prod(2:s(k))/ ... 8Waic&lX~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *(*XNd||
prod(2:((n(j)+m_abs(j))/2-s(k))); i2Gh!5]f
idx = (pows(k)==rpowers); .M\0+,%/
y(:,j) = y(:,j) + p*rpowern(:,idx); ! -gOqo
end *K=me/
3
hJ V*
if isnorm &gm/@_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); % o0.8qVJi
end ,76nDXy`
end
@|~D?&<\
% END: Compute the Zernike Polynomials ;wr]_@<~
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )#cGePA
ou&7v<)x4
% Compute the Zernike functions: Z:MU5(Te
% ------------------------------ 3Q+THg3~?
idx_pos = m>0; `&/ zOMp
idx_neg = m<0; ~x+24/qT
f^XfI H_#
z = y; GwlAEh P
if any(idx_pos) pM@0>DVi
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); W}oAgUd
end rMUQh~a/
if any(idx_neg) Wuji'sxTs
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *:,7
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end LZ~$=<
1FC1*7A[
% EOF zernfun