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 8&B{bS
function z = zernfun(n,m,r,theta,nflag) ^2XoYgv
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. b>?X8)f2e
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h$y1"!N(
% and angular frequency M, evaluated at positions (R,THETA) on the o^2.&e+dQ
% unit circle. N is a vector of positive integers (including 0), and OP{ d(~+
% M is a vector with the same number of elements as N. Each element sLPFeibof5
% k of M must be a positive integer, with possible values M(k) = -N(k) IKH#[jW'IB
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }>fL{};Z"
% and THETA is a vector of angles. R and THETA must have the same |{<g-)
% length. The output Z is a matrix with one column for every (N,M) 8g^OXZ
% pair, and one row for every (R,THETA) pair. qbpvTTF
% 1vu=2|QN
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %#Fd0L
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 0(h *<g:
% with delta(m,0) the Kronecker delta, is chosen so that the integral |&o%c/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Jx(%t<2
% and theta=0 to theta=2*pi) is unity. For the non-normalized 3T%WfS+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. OANn!nZ.
% K>"M#T
% The Zernike functions are an orthogonal basis on the unit circle. _z#zF[%
% They are used in disciplines such as astronomy, optics, and AS'a'x>8>,
% optometry to describe functions on a circular domain. x/R|i%u-s
% 8it|yK.G@&
% The following table lists the first 15 Zernike functions. qJKD|=_
% P10`X&
% n m Zernike function Normalization O\-cLI<h2
% -------------------------------------------------- dt<PZ.
% 0 0 1 1 n@Y`g{{e~
% 1 1 r * cos(theta) 2 %HpTQ
% 1 -1 r * sin(theta) 2 ;a*i*{\Rm
% 2 -2 r^2 * cos(2*theta) sqrt(6) J+kxb"#d
% 2 0 (2*r^2 - 1) sqrt(3) [89#8|+
% 2 2 r^2 * sin(2*theta) sqrt(6) 'cu(
Sd}
% 3 -3 r^3 * cos(3*theta) sqrt(8)
3_+-t5
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) o'?Y0Wt
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) -H#{[M8xX
% 3 3 r^3 * sin(3*theta) sqrt(8) &1{RuV&t
% 4 -4 r^4 * cos(4*theta) sqrt(10) Nj@k|_1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ==l p\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) .TSj8,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;Z%ysLA
% 4 4 r^4 * sin(4*theta) sqrt(10) ?TLMoqmXM{
% -------------------------------------------------- _A;jtS)SY
% D
N GNc
% Example 1: nxA Y]Q
% u yzc"di
% % Display the Zernike function Z(n=5,m=1) 5M;fh)fT
% x = -1:0.01:1; ck){N?y
% [X,Y] = meshgrid(x,x); 4t|ril``]
% [theta,r] = cart2pol(X,Y); pJ ;J>7Gt
% idx = r<=1; '(7]jug
% z = nan(size(X)); D\jRF-z
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m>yk4@a
% figure `_N8AA
% pcolor(x,x,z), shading interp @(,k%84z
% axis square, colorbar VrD?[&2pE
% title('Zernike function Z_5^1(r,\theta)') ?54=TA|5`F
% #KF:(2
% Example 2: &jT>)MXPu
% R#"kh/M
% % Display the first 10 Zernike functions A|,\}9)4X[
% x = -1:0.01:1; ,2qJXMg"=$
% [X,Y] = meshgrid(x,x); ;O}%_ef@
% [theta,r] = cart2pol(X,Y); q&B'peT
% idx = r<=1; Zrr3='^s
% z = nan(size(X)); ZT5t~5W
% n = [0 1 1 2 2 2 3 3 3 3]; u-=S_e
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; G|Yw
a=
% Nplot = [4 10 12 16 18 20 22 24 26 28]; d+[yW7%J
% y = zernfun(n,m,r(idx),theta(idx)); v7&e,:r2E@
% figure('Units','normalized') tKjPLi71
% for k = 1:10 3;zJ\a.+
% z(idx) = y(:,k); sU^2I v\%
% subplot(4,7,Nplot(k)) UeIu
-[R
% pcolor(x,x,z), shading interp hPE#l?H@A
% set(gca,'XTick',[],'YTick',[]) Ok/~E
% axis square m\(4y Gj
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #UG| \}Lp
% end /pan{.< k
% E{[c8l2B
% See also ZERNPOL, ZERNFUN2. s^TF+d?B
};o6|e:2E
% Paul Fricker 11/13/2006 zm-j FY ?
TRL4r_
zmQ V6o=k
% Check and prepare the inputs:
({zt=}r,
% ----------------------------- s3HwBA
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }{[H@uhjH
error('zernfun:NMvectors','N and M must be vectors.') M=HW2xn
end @vh3S+=M
^W5rL@h_
if length(n)~=length(m) s-Q-1lKV,
error('zernfun:NMlength','N and M must be the same length.') XaW@CW
end $qYtN`b,
]:(>r&'
n = n(:); FY)v rM*yh
m = m(:); Ir|Q2$W2^c
if any(mod(n-m,2)) :~3sW< PR
error('zernfun:NMmultiplesof2', ... <"{Lv)4
'All N and M must differ by multiples of 2 (including 0).') L MC-1
end pg1o@^OuL
TS^(<+'
if any(m>n) H=?v$!
i
error('zernfun:MlessthanN', ... AR\>P
'Each M must be less than or equal to its corresponding N.') W"? |O Q'
end mq`N&ABO!K
"(PJh\S>S
if any( r>1 | r<0 ) I~\j%zD
error('zernfun:Rlessthan1','All R must be between 0 and 1.') WCA`34(
end gRIRc4p
IzF7W?k
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;X<#y2`
error('zernfun:RTHvector','R and THETA must be vectors.') 2hdi)C,7Y
end qUA&XUJ
Vh$~]>t:f
r = r(:); ?`V%[~4_I
theta = theta(:); E#JDbV1AC
length_r = length(r); rVd (H
if length_r~=length(theta) 3Wxl7"!x m
error('zernfun:RTHlength', ... "2;$?*hO#
'The number of R- and THETA-values must be equal.') b)J(0,9`G"
end O9wZx%<
3.U5Each-
% Check normalization: `=Pn{JaD
% -------------------- I~y[8
if nargin==5 && ischar(nflag) u4bPj2N8I
isnorm = strcmpi(nflag,'norm'); 7GY[l3arxv
if ~isnorm zk=5uKcPE
error('zernfun:normalization','Unrecognized normalization flag.') nF0$
end =;!C7VS
else (`x6QiG!
isnorm = false; CT+pkNC
end |B<+Y<)f^
&?YbAo_K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2LGeRw
% Compute the Zernike Polynomials 9Xo'U;J
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2#~5[PtP^
N(q%|h<Z/=
% Determine the required powers of r: :$."x
'
% ----------------------------------- Ug*:o d
m_abs = abs(m); 0^nnR7
rpowers = []; pqFgi_2m
for j = 1:length(n) |0:<Z(
rpowers = [rpowers m_abs(j):2:n(j)]; D@*<p h=
end 5jD2%"YUV
rpowers = unique(rpowers); :"7V,UP
@
o7<pI8\
% Pre-compute the values of r raised to the required powers, `=0}+
% and compile them in a matrix: gd^1c}UZX
% ----------------------------- |_/q0#"
if rpowers(1)==0 Zy _A3m{
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }eb}oK
rpowern = cat(2,rpowern{:}); iIji[>qz
rpowern = [ones(length_r,1) rpowern]; fiqeXE?E
else .vYU4g]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?RJ
)u
rpowern = cat(2,rpowern{:}); L^uO.eI"m
end PCDsj_e
LPX@oh a
% Compute the values of the polynomials: zC#[
% -------------------------------------- 3Xyu`zS&
y = zeros(length_r,length(n)); fBBNP)
for j = 1:length(n) Gh}sk-Xk=
s = 0:(n(j)-m_abs(j))/2; .)~IoIW=
pows = n(j):-2:m_abs(j); 37Ux2t
for k = length(s):-1:1 AeR3wua
p = (1-2*mod(s(k),2))* ... FB-?{78~
prod(2:(n(j)-s(k)))/ ... `K37&b