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 z$#q'+$
function z = zernfun(n,m,r,theta,nflag) NXwthc3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. c3S}(8g5.
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H&$L1CrdL
% and angular frequency M, evaluated at positions (R,THETA) on the +qN}oyL
% unit circle. N is a vector of positive integers (including 0), and ~SKV%
% M is a vector with the same number of elements as N. Each element eBUexxBY
% k of M must be a positive integer, with possible values M(k) = -N(k) 0PfjD
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, c*>8VW>
% and THETA is a vector of angles. R and THETA must have the same 9]u=b\fzZ
% length. The output Z is a matrix with one column for every (N,M)
=K#5I<x
% pair, and one row for every (R,THETA) pair. 2HA-q),6
% =|}_ASbzw
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zG
IxmJ.
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c5 ^CWk K
% with delta(m,0) the Kronecker delta, is chosen so that the integral HL8onNq
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <Zb~tYp
% and theta=0 to theta=2*pi) is unity. For the non-normalized ~PaEhj&8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Os)jfKn2
% 4gR;,%E\TO
% The Zernike functions are an orthogonal basis on the unit circle. j
p"hbV
% They are used in disciplines such as astronomy, optics, and zx#HyO[a
% optometry to describe functions on a circular domain. exW|c~|m{A
% G_ -8*.
% The following table lists the first 15 Zernike functions. CG[2
% B3AWJ1o
% n m Zernike function Normalization 9w)W| 9
% -------------------------------------------------- sej$$m R
% 0 0 1 1 /)+V(Jlu
% 1 1 r * cos(theta) 2 rXh*nC
% 1 -1 r * sin(theta) 2 +aY]?]
% 2 -2 r^2 * cos(2*theta) sqrt(6) d76nyQKK
% 2 0 (2*r^2 - 1) sqrt(3) RIm8PV;N
% 2 2 r^2 * sin(2*theta) sqrt(6) -eE r|Gs)
% 3 -3 r^3 * cos(3*theta) sqrt(8) $U/|+*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) %XC3V7
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) )6!ji]c
N
% 3 3 r^3 * sin(3*theta) sqrt(8) *F:)S"3_~e
% 4 -4 r^4 * cos(4*theta) sqrt(10) T]_]{%z
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4Tdp;n\F
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 0<3->uK
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {s7
3(B"
% 4 4 r^4 * sin(4*theta) sqrt(10) "
""k}M2A
% -------------------------------------------------- c1Rn1M,2k
% i)!2DXn
% Example 1: qr@<'wp/
% s~p(59
% % Display the Zernike function Z(n=5,m=1) SSQB1c
% x = -1:0.01:1; y2`},
% [X,Y] = meshgrid(x,x); c0ue[tb
% [theta,r] = cart2pol(X,Y); <5 )F9.$
% idx = r<=1; &7J-m4BI
% z = nan(size(X)); m7#v2:OD+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Of}dsav
% figure 9$q35e
% pcolor(x,x,z), shading interp #c%FpR4
% axis square, colorbar fxQ4kiI
% title('Zernike function Z_5^1(r,\theta)') VbI$#;:[7
% ~T^,5Tz1j
% Example 2: koojF|H>
% 4JO[yN
% % Display the first 10 Zernike functions 14pyHMOR
% x = -1:0.01:1; xNd p]u
% [X,Y] = meshgrid(x,x); gyz_$T@x
% [theta,r] = cart2pol(X,Y); ;F;Vm$
% idx = r<=1; 0-Ga2Go9
% z = nan(size(X)); &cp
`? k
% n = [0 1 1 2 2 2 3 3 3 3];
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; LrH"d
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Y@y"bjK \
% y = zernfun(n,m,r(idx),theta(idx)); Di"Tv<RlQ
% figure('Units','normalized') ucYweXsO3
% for k = 1:10 Ie]k/qw+ Y
% z(idx) = y(:,k); 5AbY 59
% subplot(4,7,Nplot(k)) r_hs_n!6
% pcolor(x,x,z), shading interp B,fVNpqo
% set(gca,'XTick',[],'YTick',[]) GIvl|
% axis square 6r.#/' "
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) XI+GWNAmJ
% end -A,UqEt
% oZ_,WwnE
% See also ZERNPOL, ZERNFUN2. g#q7~#9
/!'Png0!
% Paul Fricker 11/13/2006 8ZF!}kb0F
Ea)=K'Pz
Cq -URih
% Check and prepare the inputs:
6DG%pF,
% ----------------------------- A<y]D.Z"
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7<ZGNxZ~
error('zernfun:NMvectors','N and M must be vectors.') cE^Ljk
end P0/Ctke;
MCAWn
H
if length(n)~=length(m) D<gd)
error('zernfun:NMlength','N and M must be the same length.') 9H/C(Vo
end ^;sE)L6
H0f] Swh0a
n = n(:); . {vMn0c
m = m(:); ?PYZW5
if any(mod(n-m,2)) mX%T"_^
error('zernfun:NMmultiplesof2', ... TQtHU6
'All N and M must differ by multiples of 2 (including 0).') Iqci}G%r
end Nwo*tb:
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if any(m>n) ..R JHa6B
error('zernfun:MlessthanN', ... 3Rhoul[S
'Each M must be less than or equal to its corresponding N.') n/{ pQ&B
end ,e^~(ITaq
8,IQ6Or|-2
if any( r>1 | r<0 ) Ob/i_
error('zernfun:Rlessthan1','All R must be between 0 and 1.') +Ks 3
end cw,|,uXq
6
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fRb
error('zernfun:RTHvector','R and THETA must be vectors.') o:B?hr'\
end v|KGzQx$.*
;H3~r^>c
r = r(:); rd;E /:`5
theta = theta(:); Z2 Vri
length_r = length(r); :Q,~Nw>
if length_r~=length(theta) au]W*;x
error('zernfun:RTHlength', ... azzG
'The number of R- and THETA-values must be equal.') ma xpR>7`j
end 5IA3\G}+
1gnLKf c
% Check normalization: P:WxhO/
% -------------------- RG=i74a
if nargin==5 && ischar(nflag) $o.;}
isnorm = strcmpi(nflag,'norm'); )gD2wk(
if ~isnorm [Op^l%BC
error('zernfun:normalization','Unrecognized normalization flag.') 2*< PmKI
end G?, "AA;
else [&IcIZ
isnorm = false; hE
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end [%P[ x]-
A)gSOC{3F)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e _(';Lk
% Compute the Zernike Polynomials Qp7F3,/#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j"jQiL_*
LqXVi80
% Determine the required powers of r: iUFG!,+d
% ----------------------------------- Ljiw9*ZI
m_abs = abs(m); g{
;OgS3>
rpowers = []; /6F\]JwU
for j = 1:length(n) )w5!'W4Z8
rpowers = [rpowers m_abs(j):2:n(j)]; NHQoP&OG
end &grvlK
rpowers = unique(rpowers); .2U3_1dX
0176
% Pre-compute the values of r raised to the required powers, Mnk-"d
% and compile them in a matrix: `e`DSl D>
% ----------------------------- `Z#':0Z
if rpowers(1)==0 .'. bokl/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zC*dJXt@
rpowern = cat(2,rpowern{:}); YNl".c
rpowern = [ones(length_r,1) rpowern]; K2\)9
else H DD)AM&p
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); K}MlC}oIt
rpowern = cat(2,rpowern{:}); `DE_<l
end CbXSJDs
x3(
->?)D
% Compute the values of the polynomials: L7n G5i
% -------------------------------------- tSnsjd<6.
y = zeros(length_r,length(n)); cW_l |
for j = 1:length(n) (74y2U6
s = 0:(n(j)-m_abs(j))/2; GY %$7
pows = n(j):-2:m_abs(j); :>0,MO.^~K
for k = length(s):-1:1 .XkD2~;
p = (1-2*mod(s(k),2))* ... *wsZ aQ
prod(2:(n(j)-s(k)))/ ... u.G aMl4 (
prod(2:s(k))/ ... p] N/]2rR
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4"3.7.<Q`
prod(2:((n(j)+m_abs(j))/2-s(k))); ir>S\VT4
idx = (pows(k)==rpowers); !G6h~`[
y(:,j) = y(:,j) + p*rpowern(:,idx); s|:1z"q
end kma>'P`G
fFoZ!H
if isnorm 8^D1u`
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xX9snSGz
end n(|n=P:o
end OSLZ7B^
% END: Compute the Zernike Polynomials h@'CmIZc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &c