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 1.SkIu%
function z = zernfun(n,m,r,theta,nflag) wq4nMY:#
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. B#tdLv"I
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 67J*&5? |
% and angular frequency M, evaluated at positions (R,THETA) on the HR3_@^<7
% unit circle. N is a vector of positive integers (including 0), and n=`w9qajd
% M is a vector with the same number of elements as N. Each element jNy?[
)
% k of M must be a positive integer, with possible values M(k) = -N(k) lug}
Uj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, !*P&Eat
% and THETA is a vector of angles. R and THETA must have the same |5xz l
% length. The output Z is a matrix with one column for every (N,M) kUHie
% pair, and one row for every (R,THETA) pair. _
K/swT{f
% %yaG,;>U
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PZ34 *q
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 6.Bh3p
% with delta(m,0) the Kronecker delta, is chosen so that the integral vF>gU_gz.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, yL"i
% and theta=0 to theta=2*pi) is unity. For the non-normalized j??tmo
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. m.V,I}J.q
% g 2'x#%ET
% The Zernike functions are an orthogonal basis on the unit circle. b|ZLX:
% They are used in disciplines such as astronomy, optics, and !"! ii$@
% optometry to describe functions on a circular domain. ek[kq[U9
% 6;JP76PD
% The following table lists the first 15 Zernike functions. y`b\;kd
% >38
Lt\
% n m Zernike function Normalization C|6{fd4?
% -------------------------------------------------- pGGV\zD^
% 0 0 1 1 Dq`~XS*
% 1 1 r * cos(theta) 2 '\L0xw4
% 1 -1 r * sin(theta) 2 ny`(f,)u*
% 2 -2 r^2 * cos(2*theta) sqrt(6) ZT9IMihV
% 2 0 (2*r^2 - 1) sqrt(3) #` +]{4hR
% 2 2 r^2 * sin(2*theta) sqrt(6) aFG3tuaKrQ
% 3 -3 r^3 * cos(3*theta) sqrt(8) _j 5N=I{U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) NV#')+Ba
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) rBevVc![
% 3 3 r^3 * sin(3*theta) sqrt(8) aQmfrx
% 4 -4 r^4 * cos(4*theta) sqrt(10) WW3
B
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C*O
,rm}
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Y*\6o7
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6z1\a
% 4 4 r^4 * sin(4*theta) sqrt(10) C|$L6n>DR6
% -------------------------------------------------- \[T{M!s
% fN0bIE
Y
% Example 1: \ 522,n`
% -,/3"}<^78
% % Display the Zernike function Z(n=5,m=1) qsvpW%?aE
% x = -1:0.01:1; e;;):\p4
% [X,Y] = meshgrid(x,x); \c68n
% [theta,r] = cart2pol(X,Y); !a4cjc(
% idx = r<=1; bqjr0A7{
% z = nan(size(X)); 8{@`kyy|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); bx7\QU+
% figure }Eb]9c\
% pcolor(x,x,z), shading interp V{FE [v_
% axis square, colorbar bpnv &EG
% title('Zernike function Z_5^1(r,\theta)') :Q=z=`*2w
% SJOmeN}4)
% Example 2: fwH`}<o
% tO4):i1
% % Display the first 10 Zernike functions JE9>8+
% x = -1:0.01:1; Ym:{Mm=ud
% [X,Y] = meshgrid(x,x); Nor`c+,4
% [theta,r] = cart2pol(X,Y); H1C%o0CPY
% idx = r<=1; Dh?vU~v(6
% z = nan(size(X)); enPLaiJ'|q
% n = [0 1 1 2 2 2 3 3 3 3]; ,,}sK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; K{N%kk%F
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Tr$i=
M
% y = zernfun(n,m,r(idx),theta(idx)); `1$y( w]
% figure('Units','normalized') +h|K[=l\
% for k = 1:10 +
lP5XY{
% z(idx) = y(:,k); EFwL.'Fh
% subplot(4,7,Nplot(k)) bk0Y
% pcolor(x,x,z), shading interp T|!D>l'
% set(gca,'XTick',[],'YTick',[]) [='p!7z
% axis square 9,w}Xe=C
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) r/^tzH's
% end *i%.{ YH
% mw ?{LT
% See also ZERNPOL, ZERNFUN2. p;F2z;#
e"PMvQ
% Paul Fricker 11/13/2006 -}< d(c
'1]+8E
`Z
fMyE}z
% Check and prepare the inputs: }U(\~
=D
% ----------------------------- \U Ax(;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) jjX'_E
error('zernfun:NMvectors','N and M must be vectors.') 90?,-6
end erXy>H[;
%
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if length(n)~=length(m) z^gi[
mi
error('zernfun:NMlength','N and M must be the same length.') ~~U<
end L)1C'8).
U%h7h`=F?
n = n(:); A"0wvk)UcY
m = m(:); jzMhJ
if any(mod(n-m,2)) \Oz,Qzr|
error('zernfun:NMmultiplesof2', ... v;Swo("
'All N and M must differ by multiples of 2 (including 0).') Lr wINVa
end XynU/Go,
~Vwk:+):
if any(m>n) NoJUx['6
error('zernfun:MlessthanN', ... m**0rpA
'Each M must be less than or equal to its corresponding N.') y-%nJD$
end ]c5DOv&
(rAiDRQ[
if any( r>1 | r<0 ) ss/h[4h4h
error('zernfun:Rlessthan1','All R must be between 0 and 1.') l_bL,-|E8
end N?\bBt@
(%6(5,
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #"hJpyW 4V
error('zernfun:RTHvector','R and THETA must be vectors.') -QN1oK@\mE
end t3pZjdLJd
{ms,q_Zr
r = r(:); ,Y$F7&
theta = theta(:); C:rRK*
length_r = length(r); sKe,
if length_r~=length(theta) +{5JDyh0
error('zernfun:RTHlength', ... '`9%'f)
'The number of R- and THETA-values must be equal.') gW'P`Oxw
end ~g*Y,
Y
<9ePi9D(
% Check normalization: Y||yzJdC
% -------------------- wTB)v !
if nargin==5 && ischar(nflag) 3w
t:5
Im
isnorm = strcmpi(nflag,'norm'); AQB1gzE
if ~isnorm |sA4:Aq
error('zernfun:normalization','Unrecognized normalization flag.') Tld1P69(
end &7$,<9.
else XyvZ&d6(d
isnorm = false; m5X3{[a:
end yT[Lzv#
aUKh})B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ov?.:M
% Compute the Zernike Polynomials '.]e._T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dNOX&$/=
I~d#p ]>
% Determine the required powers of r: "L9C
% ----------------------------------- KYnW7|*
m_abs = abs(m); #=`FM:WH
rpowers = []; nu#aa#ex>
for j = 1:length(n) eFt\D\XOW
rpowers = [rpowers m_abs(j):2:n(j)]; @*CAn(@#N
end =@Q#dDnFu%
rpowers = unique(rpowers); >(IITt
z0T`5NG@
% Pre-compute the values of r raised to the required powers, -@YVe:$%b
% and compile them in a matrix: 4C l,Iw/;
% ----------------------------- =#OHxM
if rpowers(1)==0 \Ku9"x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +L^A:}L(
rpowern = cat(2,rpowern{:}); pi^^L@@d
rpowern = [ones(length_r,1) rpowern]; R2Twm!1
else g,00'z_D
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }s`jl``PM
rpowern = cat(2,rpowern{:}); C_;HaQiu
end Am>_4
Zk~nB}Xw
% Compute the values of the polynomials:
80{#bb
% -------------------------------------- P]!LN\[
y = zeros(length_r,length(n)); k)N2 +/
for j = 1:length(n) y3&Tv
s = 0:(n(j)-m_abs(j))/2; a"`g"ZRx
pows = n(j):-2:m_abs(j); =giM@MV
for k = length(s):-1:1 [ea6dv4p
p = (1-2*mod(s(k),2))* ... S% JNxT7'
prod(2:(n(j)-s(k)))/ ... 03X<x|
prod(2:s(k))/ ... s(1_:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... SRA|7g}7W
prod(2:((n(j)+m_abs(j))/2-s(k))); c*y$bf<
idx = (pows(k)==rpowers); 2x)0?N[$O
y(:,j) = y(:,j) + p*rpowern(:,idx); NWo7wVwc/c
end * 23m-
xT_fr,P
if isnorm O, bfdc[g4
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1$='`@8I
end r[.zLXgK
end _Vdb?
% END: Compute the Zernike Polynomials .jU|gf:x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B(4:_j\2
Fpj6Atk
% Compute the Zernike functions: OoAr%
% ------------------------------
o9U0kI=W
idx_pos = m>0; <.PPs:{8#
idx_neg = m<0; 8w{#R{w
eh({K;>
z = y; Z$OF|ZZQ
if any(idx_pos) K#9(|2J%
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `-72>F ;T
end &=s|
if any(idx_neg) E1Ru)k{B
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &%f ]-=~
end s${T*)S@G
,xtKPA
% EOF zernfun