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 ;Aiuy{<
function z = zernfun(n,m,r,theta,nflag) [kgT"?w=
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7am ._K
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9]BpP0f\
% and angular frequency M, evaluated at positions (R,THETA) on the ~;,]/'O
% unit circle. N is a vector of positive integers (including 0), and ~d ~$fR
% M is a vector with the same number of elements as N. Each element 3'O+
% k of M must be a positive integer, with possible values M(k) = -N(k) PkQu N;a
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3k5OYUk
% and THETA is a vector of angles. R and THETA must have the same {*As-Y:'F
% length. The output Z is a matrix with one column for every (N,M) Vp\BNq_!s
% pair, and one row for every (R,THETA) pair. Ec[=~>;n{l
% "0+_P{w+
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "{&\ nt
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 0O+s3#"?@
% with delta(m,0) the Kronecker delta, is chosen so that the integral gzvEy^X
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bT*MJ7VVm
% and theta=0 to theta=2*pi) is unity. For the non-normalized P*T'R
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 97e fWYj
% zht^gOs
% The Zernike functions are an orthogonal basis on the unit circle. \CM(
% They are used in disciplines such as astronomy, optics, and K0yTHX?(.
% optometry to describe functions on a circular domain.
]nhLv!Co
% 7 w_`<b6
% The following table lists the first 15 Zernike functions. K!"[,=u_
% FJKt5}`8
% n m Zernike function Normalization c~b[_J)
% -------------------------------------------------- ~d^+yR-
% 0 0 1 1 WZ'8{XY8
% 1 1 r * cos(theta) 2 p@/!+$^{
% 1 -1 r * sin(theta) 2 a Umcs!@
% 2 -2 r^2 * cos(2*theta) sqrt(6) NQ !t `
% 2 0 (2*r^2 - 1) sqrt(3)
FAJ\9
% 2 2 r^2 * sin(2*theta) sqrt(6) C;}~C:aJ
% 3 -3 r^3 * cos(3*theta) sqrt(8) THWT\3~,
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) U_m<W$"HF
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 9kuL1tcY
% 3 3 r^3 * sin(3*theta) sqrt(8) U")~bU
% 4 -4 r^4 * cos(4*theta) sqrt(10) 7gfNe kr~W
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `MlQPLH
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'ADt<m_$
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 49^;T;'v
% 4 4 r^4 * sin(4*theta) sqrt(10) BJ<hP9#
% -------------------------------------------------- ` QXO+'j4
% JGX E{FT
% Example 1: 2PE|4zG
% @zB {Ig
% % Display the Zernike function Z(n=5,m=1) ~t n*y4uK
% x = -1:0.01:1; }RYr)
% [X,Y] = meshgrid(x,x); t@QaxZIlt;
% [theta,r] = cart2pol(X,Y); )7Gm<r
% idx = r<=1; wAkpk&R
% z = nan(size(X)); k q8:h
% z(idx) = zernfun(5,1,r(idx),theta(idx)); r@f8-!{s2h
% figure %RG kXOgp
% pcolor(x,x,z), shading interp xmb]L:4F
% axis square, colorbar RZ:Yu
% title('Zernike function Z_5^1(r,\theta)') fQ=Yf ?b
% "yXKu)_
% Example 2: g2JNa?z
% <w`
R;
% % Display the first 10 Zernike functions d^mw&F)S
% x = -1:0.01:1; ;"-(QE?Mv
% [X,Y] = meshgrid(x,x); f)l:^/WP+
% [theta,r] = cart2pol(X,Y); UX;?~X
% idx = r<=1; Ij` %'/J
% z = nan(size(X)); S3EY9:^C
% n = [0 1 1 2 2 2 3 3 3 3]; 8{#WF#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; V$VqYy9 *
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ynvU$}w ~'
% y = zernfun(n,m,r(idx),theta(idx)); >N62t9Ll[
% figure('Units','normalized') z6]dF"N
% for k = 1:10 UBzX%:A
% z(idx) = y(:,k); &YGd!Q
% subplot(4,7,Nplot(k)) G|Rsj{2'
% pcolor(x,x,z), shading interp z85%2Apd
% set(gca,'XTick',[],'YTick',[]) +%7v#CY
&
% axis square M(KsLu1
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @)1>ba
% end 7n9&@D3:P
% f_Ma~'3
% See also ZERNPOL, ZERNFUN2. :JH#*5%gQ:
y^zII5|s
% Paul Fricker 11/13/2006 <k!M+}a 9V
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1"YN{Ut;G
% Check and prepare the inputs: X]8(_[Y
% ----------------------------- JFH3)Q
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FeoI+KA
error('zernfun:NMvectors','N and M must be vectors.') r&oR|-2hRk
end OB`(,m#
c.dk4v%Y5
if length(n)~=length(m) L[lX?g?Ob
error('zernfun:NMlength','N and M must be the same length.') U$v|c%6
end
I{tY;b'w
]6L;
n = n(:); N;4bEcWjp
m = m(:); p.6C.2q~s]
if any(mod(n-m,2)) Swz{5 J2C
error('zernfun:NMmultiplesof2', ... )UbPG`x8
'All N and M must differ by multiples of 2 (including 0).') $9+|_[ ]v.
end 39to5s,
H
xs'VK*
if any(m>n) ]xC#XYE:dy
error('zernfun:MlessthanN', ... WJWi'|C4
'Each M must be less than or equal to its corresponding N.') \~m\pf?
end uP, iGA
${m;x: '
if any( r>1 | r<0 ) q\s"B.(G"
error('zernfun:Rlessthan1','All R must be between 0 and 1.') |_."U9!Z^
end VzfaUAIZl
[ )3rc}:1
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) n2}(Pt.
error('zernfun:RTHvector','R and THETA must be vectors.') K8#MQR2@
end $`j%z@[g
jq%%|J.x
r = r(:); Em)U`"j/9
theta = theta(:); } I>6 8dS[
length_r = length(r); $inlI_
if length_r~=length(theta) g)$/'RB
error('zernfun:RTHlength', ... 6&|hpp#[
'The number of R- and THETA-values must be equal.') #1*#3p9UL
end 4>
k"$l/:
yq. <,b=87
% Check normalization: ICck 0S!
% -------------------- RO+ jVY~H-
if nargin==5 && ischar(nflag) ]%M&pc3U
isnorm = strcmpi(nflag,'norm'); JfD-CoQS'
if ~isnorm e}dGK=`
error('zernfun:normalization','Unrecognized normalization flag.') .3
>"qv
end pwvzs`[;
else F>Pr`T?>
isnorm = false; a-e_ q
end &!P' M
@)#EZQi x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RW~!)^
% Compute the Zernike Polynomials .~$!BWP
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $%BI8_
nQGl]2
% Determine the required powers of r: Cj%n?-
% ----------------------------------- e!W U
m_abs = abs(m); cWtuI(.
rpowers = []; [Ef6@
for j = 1:length(n) mR|L'[l
rpowers = [rpowers m_abs(j):2:n(j)]; 9?X8H1
end :@uIEvD?
rpowers = unique(rpowers); >``sM=W at
9xi nX-x;n
% Pre-compute the values of r raised to the required powers, r7)qr%n
% and compile them in a matrix: QyghNImp
% ----------------------------- IR2=dQS
if rpowers(1)==0 = N&5]Z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); L4DT*(;!E
rpowern = cat(2,rpowern{:}); Vv54;Js9
rpowern = [ones(length_r,1) rpowern]; OZc4 -5
else Ff{,zfN+3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l1bkhA b
rpowern = cat(2,rpowern{:}); :KmnwYm
end 44NMof8N
HQvJ*U4++
% Compute the values of the polynomials: GO?hB4 9T
% -------------------------------------- xi51,y+(5
y = zeros(length_r,length(n)); 3
,zW6 -}
for j = 1:length(n) 4#CHX^De
s = 0:(n(j)-m_abs(j))/2; X+1Mv
pows = n(j):-2:m_abs(j); NSa6\.W)
for k = length(s):-1:1 fB80&G9
p = (1-2*mod(s(k),2))* ... ]_BH"ng}
prod(2:(n(j)-s(k)))/ ... ZDG~tCh=@
prod(2:s(k))/ ... yky%+@2q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e2e!"kEF
prod(2:((n(j)+m_abs(j))/2-s(k))); G9^xv
idx = (pows(k)==rpowers); IRGcE&m
y(:,j) = y(:,j) + p*rpowern(:,idx); :8K}e]!c1
end q<j9l'dHG
\TZSn1isZX
if isnorm @9eN\b%I^H
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2x>7>;>
end U9ZuD40\
end M8Vc5
% END: Compute the Zernike Polynomials 6Df*wi!jI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Jz}`-fU`
<UF0Xc&X'
% Compute the Zernike functions: Xp] jF^5
% ------------------------------ nY7gST
idx_pos = m>0; QChncIqc
idx_neg = m<0; Esu{c9,
ta6>St7.
z = y; jST4O"DjM
if any(idx_pos) eTFep^[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O6/:J#X%
end oYdE s&qq
if any(idx_neg) $*VZa3B\
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T/A2Y+@N;
end _p>F43%p
r<'DS9m
% EOF zernfun