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 tEpIyC
function z = zernfun(n,m,r,theta,nflag) N (:E K
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. >o0&:h|>$'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 97:t29N
% and angular frequency M, evaluated at positions (R,THETA) on the X~IRpzC
% unit circle. N is a vector of positive integers (including 0), and w~cq%%
% M is a vector with the same number of elements as N. Each element b@{%qh,C
% k of M must be a positive integer, with possible values M(k) = -N(k) -z>Z0viA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ^rxfNcU7
% and THETA is a vector of angles. R and THETA must have the same }"g21-T^
% length. The output Z is a matrix with one column for every (N,M) 1)P<cNj
% pair, and one row for every (R,THETA) pair. []6ShcqJ[v
% FcA)RsMI*
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike s/W!6JX4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !%Z)eO~Z
% with delta(m,0) the Kronecker delta, is chosen so that the integral =:CGl
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, rA1zyZlz
% and theta=0 to theta=2*pi) is unity. For the non-normalized Q%X:5G?
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. eCPKpVhP
%
\
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% The Zernike functions are an orthogonal basis on the unit circle. o1p$9PL\:
% They are used in disciplines such as astronomy, optics, and :$GL.n-?
% optometry to describe functions on a circular domain. ~_f
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% ecSdU>
% The following table lists the first 15 Zernike functions. Hz!U_?
% @Le ^- v4
% n m Zernike function Normalization #um1?V
% -------------------------------------------------- -Z/6;2Q
% 0 0 1 1 Y7b,td1
% 1 1 r * cos(theta) 2 s$DT.cvO
% 1 -1 r * sin(theta) 2 Gct&}]3pm
% 2 -2 r^2 * cos(2*theta) sqrt(6) \U<F\i
% 2 0 (2*r^2 - 1) sqrt(3) @2%VU#!m
% 2 2 r^2 * sin(2*theta) sqrt(6) 8IT_mjj
% 3 -3 r^3 * cos(3*theta) sqrt(8) C,VqT6E<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~q#[5l(r8
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 6>LQGO
% 3 3 r^3 * sin(3*theta) sqrt(8) 6/V{>MTZg
% 4 -4 r^4 * cos(4*theta) sqrt(10) ~'Qpf 8)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kERaY9L\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ZhJ|ZvJ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "$,}|T?Y`
% 4 4 r^4 * sin(4*theta) sqrt(10) om*tdG
% -------------------------------------------------- KOAz-h@6
% "PDSqYA
% Example 1: )z4kP09
% KH@) +Rj
% % Display the Zernike function Z(n=5,m=1) yoA*\V
% x = -1:0.01:1; qhn&;{{
% [X,Y] = meshgrid(x,x); !w;A=
% [theta,r] = cart2pol(X,Y); 1TD&&EC
% idx = r<=1; 9bzYADLI
% z = nan(size(X)); KoQ_:`
% z(idx) = zernfun(5,1,r(idx),theta(idx));
5Ky9P z
% figure L2/<+Zw
% pcolor(x,x,z), shading interp $.3CiM}~
% axis square, colorbar v^lm8/}NO
% title('Zernike function Z_5^1(r,\theta)') %;B(_ht<-w
% WKYA9BaR
% Example 2: fXXm@tMx>
% JG+g88
% % Display the first 10 Zernike functions <+i`W7
% x = -1:0.01:1; ^&G O4u
% [X,Y] = meshgrid(x,x); zx]M/=7,V#
% [theta,r] = cart2pol(X,Y); b#\kZ/W
% idx = r<=1; ETH#IM8J
% z = nan(size(X)); B"E (Y M
% n = [0 1 1 2 2 2 3 3 3 3]; Jk6/i;4|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; >`,#%MH#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; HNHhMi`w
% y = zernfun(n,m,r(idx),theta(idx)); 1rm$@L
% figure('Units','normalized') enD C#
% for k = 1:10 UgP=k){
% z(idx) = y(:,k); BS<>gA
R;/
% subplot(4,7,Nplot(k)) aY1#K6(y
% pcolor(x,x,z), shading interp -"JE-n
% set(gca,'XTick',[],'YTick',[]) Vo9)KxR
% axis square jtVPv]
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0wE8GmG
% end C7*Yg$`{
% j"$b%|
% See also ZERNPOL, ZERNFUN2. I}Gl*@K&O
Nno={i1jk
% Paul Fricker 11/13/2006 *}WqYqOow
1
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NYV0<z@M2M
% Check and prepare the inputs: G}hkr
% ----------------------------- |sZ9/G7
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ])ZJ1QL1
error('zernfun:NMvectors','N and M must be vectors.') ^MWW,`
end {Z~VO
QX~72X=(
if length(n)~=length(m) O6/=/-?N=c
error('zernfun:NMlength','N and M must be the same length.') P@T $6%~
end qP .VK?jF|
H&K)q5~
n = n(:); &WWO13\qd
m = m(:); 6`$z*C2{
if any(mod(n-m,2)) M+&eh*:z:
error('zernfun:NMmultiplesof2', ... FUv)<rK
'All N and M must differ by multiples of 2 (including 0).') w7ABnX
end P*^UU\x'4I
GH)+yD[o
if any(m>n) oIR%{`3"I
error('zernfun:MlessthanN', ... !Q/O[6
'Each M must be less than or equal to its corresponding N.') |c+N)FB
end $HnD|_*
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if any( r>1 | r<0 ) W;x LuKIG
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,4I6Rw B.
end xx2:5
vH:+
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) C&K(({5O
error('zernfun:RTHvector','R and THETA must be vectors.') L$07u{Q
end 7^}Z%c
I
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r = r(:); O#Ax P}
theta = theta(:); HE.Dl7{
length_r = length(r); gYIYA"xN`
if length_r~=length(theta) C4d1*IQk
error('zernfun:RTHlength', ... ON=ley
'The number of R- and THETA-values must be equal.') sU3V)7"
end kR|DzB7
*xN jhR]7v
% Check normalization: )$.9WlQ
% -------------------- 8{^GC(W{]
if nargin==5 && ischar(nflag) {y%O_-C'r
isnorm = strcmpi(nflag,'norm'); 20xGj?M
if ~isnorm Xpz-@fqKdf
error('zernfun:normalization','Unrecognized normalization flag.') 8k}CR)3@C
end 5N}|VGN
else #z5?Y2t7~^
isnorm = false; .:Xe* Q
end ^O9m11
yq^$H^_O
p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {.'g!{SHp
% Compute the Zernike Polynomials QLLVOJi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xg!\C@$
?4dd|n
% Determine the required powers of r: Z?[J_[ZtR3
% ----------------------------------- 6h"?3w
m_abs = abs(m); os+wTUR^
rpowers = []; e"09b<69
for j = 1:length(n) e/l?|+m 6
rpowers = [rpowers m_abs(j):2:n(j)]; iFT3fP'> 5
end 5%$kAJZC-
rpowers = unique(rpowers); c=mFYsSv
C
/VXyl@o
% Pre-compute the values of r raised to the required powers, z@LP9+?dE
% and compile them in a matrix: 1Ee>pbd
% ----------------------------- _e^V\O>
if rpowers(1)==0 667tL(
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _$x *CP0(
rpowern = cat(2,rpowern{:}); Yhdt8[ 2
rpowern = [ones(length_r,1) rpowern]; XX;%:?n
else l )eaIOyk
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :zZM&r>
rpowern = cat(2,rpowern{:}); ?["ZEa
end jF0BWPL
4}b:..Ku
% Compute the values of the polynomials: 1%{(?uz9
% -------------------------------------- 2^j9m}`
y = zeros(length_r,length(n)); U4/$4.'NQ
for j = 1:length(n) p_N=V. w
s = 0:(n(j)-m_abs(j))/2; 0
N^V&k
pows = n(j):-2:m_abs(j); }e6:&`a xD
for k = length(s):-1:1 Lf<9GYNy>`
p = (1-2*mod(s(k),2))* ... @J)vuGS
prod(2:(n(j)-s(k)))/ ... ]5L3[A4Vu
prod(2:s(k))/ ... [C#pMLp,~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |8rJqtf +&
prod(2:((n(j)+m_abs(j))/2-s(k))); 2^+"GCo
idx = (pows(k)==rpowers); Q?q
m~wD
y(:,j) = y(:,j) + p*rpowern(:,idx); ;W"[,#2TM
end (/BkwbJyE
{hR23eE)#
if isnorm UJ&,9}L8
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); n(MEG'9}
end 8t"DQ Y-R
end h Nwb.[
% END: Compute the Zernike Polynomials &ICO{#v5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yLX\pkAt4
XsC bA8Qv
% Compute the Zernike functions: EtG)2)
% ------------------------------ -"H9 W:
idx_pos = m>0; kDQXPp
idx_neg = m<0; cke[SUH,
ivk|-C'\
z = y; S]a$w5ZP
if any(idx_pos) bL%)k61G_v
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pq`MO
.R
end +cN2 KP
if any(idx_neg) bf+2c6_BN0
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $P~ a
end '`"&RuB
~>|U %3}]
% EOF zernfun