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 b-8}TTL>
function z = zernfun(n,m,r,theta,nflag) [&(~{#}M:
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]`eP"U{
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 52,[dP,g
% and angular frequency M, evaluated at positions (R,THETA) on the 8
$qj&2 N
% unit circle. N is a vector of positive integers (including 0), and wn-1fz<d
% M is a vector with the same number of elements as N. Each element WuuF&0?8C
% k of M must be a positive integer, with possible values M(k) = -N(k) Q{[l1:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, A3mvd-k
% and THETA is a vector of angles. R and THETA must have the same <uG6!P
% length. The output Z is a matrix with one column for every (N,M) /@ww"dmqU
% pair, and one row for every (R,THETA) pair. q-hR EO
% .Gt_~x
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;mT
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !S~0T!afF
% with delta(m,0) the Kronecker delta, is chosen so that the integral xovsh\s
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vSnGPLl
% and theta=0 to theta=2*pi) is unity. For the non-normalized x^zw1e,y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QYg V[\&
% i 558&:
% The Zernike functions are an orthogonal basis on the unit circle. ;Zm-B]\
% They are used in disciplines such as astronomy, optics, and EVlj#~mV
% optometry to describe functions on a circular domain. fc&djd`FuX
% 6Ki!j<
% The following table lists the first 15 Zernike functions. (kTu6t*
% 5pT8 }?7
% n m Zernike function Normalization {E[t(Ig
% -------------------------------------------------- s(T0lul
% 0 0 1 1 Xf#+^cQ
% 1 1 r * cos(theta) 2 =PF2p'.o
% 1 -1 r * sin(theta) 2 ]ZnASlc)
% 2 -2 r^2 * cos(2*theta) sqrt(6) YK\pV'&+
% 2 0 (2*r^2 - 1) sqrt(3) Vk> &
% 2 2 r^2 * sin(2*theta) sqrt(6) O9P+S|hcY
% 3 -3 r^3 * cos(3*theta) sqrt(8) L}
"bp
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *Z$W"JP
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) '%X29B5
% 3 3 r^3 * sin(3*theta) sqrt(8) esiU._:u
% 4 -4 r^4 * cos(4*theta) sqrt(10) j{j5TvsrY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }Y^o("c(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) I_m3|VCa|t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bcq&yL'D
% 4 4 r^4 * sin(4*theta) sqrt(10)
OqWm5(u&S
% -------------------------------------------------- : *XAQb0
% g< xE}[gF
% Example 1: d_,Ql708f
% fK6[ p&
% % Display the Zernike function Z(n=5,m=1) ?b:Pl{?
% x = -1:0.01:1; >F>VlRg
% [X,Y] = meshgrid(x,x); bg!(B<!X
% [theta,r] = cart2pol(X,Y); i)$P1h
% idx = r<=1; kY?tUpM!TB
% z = nan(size(X)); * RyU*au
% z(idx) = zernfun(5,1,r(idx),theta(idx)); $ q*a}d[Q
% figure 'QQq0.
% pcolor(x,x,z), shading interp a>6D3n
W
% axis square, colorbar #mU<]O
% title('Zernike function Z_5^1(r,\theta)') Z($i+L% .
% I 12Zh7Cc:
% Example 2: 02tt.0go
% C1fd@6
% % Display the first 10 Zernike functions EDz;6Z*4N
% x = -1:0.01:1; }hsNsQ
% [X,Y] = meshgrid(x,x); Gy[anDE&
% [theta,r] = cart2pol(X,Y); c4u/tt.)
% idx = r<=1; <(@Z#%O9)
% z = nan(size(X)); {i+
o'Lw
% n = [0 1 1 2 2 2 3 3 3 3]; !u'xdV+bf
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; gD51N()s,
% Nplot = [4 10 12 16 18 20 22 24 26 28]; u]Q}jqiq"
% y = zernfun(n,m,r(idx),theta(idx)); ol41%q*
% figure('Units','normalized') MhR`
% for k = 1:10 a{L&RRJ
% z(idx) = y(:,k); I(Qz%/ Ox
% subplot(4,7,Nplot(k)) F b?^+V]9
% pcolor(x,x,z), shading interp S]ayH$w\Q
% set(gca,'XTick',[],'YTick',[]) ,oUzaEX
% axis square h=S7Z:IaM
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %W8iC%~
% end %Z4*;VwQ
% 8h0C G]
% See also ZERNPOL, ZERNFUN2. 8{=|<
HAL\j5i
% Paul Fricker 11/13/2006 ht*(@MCr<
78{9@\e"0
ii_kgqT^
% Check and prepare the inputs: "AZ|u#0P
% ----------------------------- .8Bu%Sf
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G^tazAEfo
error('zernfun:NMvectors','N and M must be vectors.') P
JATRJ1.
end xxyc^\$
Wlxmp['Bh
if length(n)~=length(m) g<(!>:h
error('zernfun:NMlength','N and M must be the same length.') wgIm{;T[u
end {f\wIZ-K A
p:TE##
n = n(:); /='0W3+o*L
m = m(:); $K!Jm7O\
if any(mod(n-m,2)) $cIaLq
error('zernfun:NMmultiplesof2', ... |,@D<
'All N and M must differ by multiples of 2 (including 0).') $1"gFg
end 1&! i:F#
R;!@
xy
if any(m>n) CV\^gTPmx
error('zernfun:MlessthanN', ... "d:rPJT)(@
'Each M must be less than or equal to its corresponding N.') 41Z@_J|&
end Cyd/HTNh<
bJetqF6n
if any( r>1 | r<0 ) :P}3cl_
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Gn=b_!
end |,p"<a!+{w
{=3A@/vM
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ij7P-5=<
error('zernfun:RTHvector','R and THETA must be vectors.') {h|<qfH
end 7tXy3-~biz
P4q5#r
r = r(:); A[uE#T^
theta = theta(:); ':fp|m)M
length_r = length(r); ru@#s2
if length_r~=length(theta) (ne[a2%>
error('zernfun:RTHlength', ... $/s"It
'The number of R- and THETA-values must be equal.') ;.Bz'Q
end 2PYn zAsl
mP&\?
% Check normalization: aaig1#a@1b
% -------------------- z'm}p
if nargin==5 && ischar(nflag) #Z1-+X8P
isnorm = strcmpi(nflag,'norm'); j{OA%G(I
if ~isnorm b'\Q/;oz>
error('zernfun:normalization','Unrecognized normalization flag.') '";#v.!
end .*x:
else ,Q56A#Y\
isnorm = false; X#t tDB
end ,_u7@Ix
Cu8mN B{H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +#MXeUX"
% Compute the Zernike Polynomials ;Y\LsmZ;F
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fr/EkL1Dl
$KYGQP
% Determine the required powers of r: A:< %>
% ----------------------------------- It[51NMal
m_abs = abs(m); ?{qUn8f2
rpowers = []; 8In\Jo$|q>
for j = 1:length(n) 4HGTgS
rpowers = [rpowers m_abs(j):2:n(j)]; 7.<jdp
end EL`|>/[J
rpowers = unique(rpowers); g8N"-j&@
,gVVYH?qR
% Pre-compute the values of r raised to the required powers, 3_)I&RM
% and compile them in a matrix: xT"V9t[f
% ----------------------------- RG{T\9]n
if rpowers(1)==0 `f; w
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;[::&qf
rpowern = cat(2,rpowern{:}); KkZx6A)$u
rpowern = [ones(length_r,1) rpowern]; 4C =W~6~
else Uw("+[ 5O0
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); LZn'+{\`
rpowern = cat(2,rpowern{:}); LG&BWs!
end TI DgIK
oRCc8&
% Compute the values of the polynomials: p-}X=O$
% -------------------------------------- Jj\4P1|' 7
y = zeros(length_r,length(n)); 3[[oAp
for j = 1:length(n) cF8
2wg
s = 0:(n(j)-m_abs(j))/2; Rlewp8?LB
pows = n(j):-2:m_abs(j); ?gMx
for k = length(s):-1:1 Z6zV 9hn
p = (1-2*mod(s(k),2))* ... J=^IS\m
prod(2:(n(j)-s(k)))/ ... VO:4wC"7
prod(2:s(k))/ ... mLuNl^)3
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J#..xJ?XRD
prod(2:((n(j)+m_abs(j))/2-s(k))); 2|>\A.I|=
idx = (pows(k)==rpowers); >}V?GK36
y(:,j) = y(:,j) + p*rpowern(:,idx); !"F;wg$
end J 6KHc^,7
L[Vk 6e
if isnorm Y6v{eWtSn
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); vN{@c(=g
end _Q5mPBO
end `DY
yK?R
% END: Compute the Zernike Polynomials qi4P(s-i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5*%Gh&)
wD9K\%jIr!
% Compute the Zernike functions: >R F|Q
% ------------------------------ EH|+S
idx_pos = m>0; ,R[$S"]!SH
idx_neg = m<0; l
;:IL\*1I
uxf,95<g)
z = y; E@SFK=`
if any(idx_pos) l53i
{o
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dQj/Sr
end W"Ip]LJ
if any(idx_neg) @)U.Dbm
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?#K.D vGJ
end LlX)xJ
a#j,0FKv
% EOF zernfun