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 qyto`n7
function z = zernfun(n,m,r,theta,nflag) #,sJd ^uI
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. (@zn[Nq
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N O7W}Z1G
% and angular frequency M, evaluated at positions (R,THETA) on the K^+B"
% unit circle. N is a vector of positive integers (including 0), and !jm
a --
% M is a vector with the same number of elements as N. Each element 4b)xW&K{
% k of M must be a positive integer, with possible values M(k) = -N(k) @)}U\=
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 8wOr`ho B
% and THETA is a vector of angles. R and THETA must have the same `?:'_Ki
% length. The output Z is a matrix with one column for every (N,M) BLRrHaX0
% pair, and one row for every (R,THETA) pair. %2.T1X%!
% :}lE@Y,R
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8cHZBM7'
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), "F^EfpcJ{9
% with delta(m,0) the Kronecker delta, is chosen so that the integral O3Uu{'=0
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, GC~::m~
% and theta=0 to theta=2*pi) is unity. For the non-normalized F]&9Lp}
"
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. j2z$kw%
% |Z<adOg
% The Zernike functions are an orthogonal basis on the unit circle. &8N\
6K=
% They are used in disciplines such as astronomy, optics, and :?,&u,8
% optometry to describe functions on a circular domain. ,F1$Of/'@\
% `JC!uc
% The following table lists the first 15 Zernike functions. WJ%b9{<
% ^m~=<4eX
% n m Zernike function Normalization JO$0Z
% -------------------------------------------------- Gfvz%%>l
% 0 0 1 1 eK`tFs,u
% 1 1 r * cos(theta) 2 *ULXJZ%
% 1 -1 r * sin(theta) 2 TS-[p d
% 2 -2 r^2 * cos(2*theta) sqrt(6) .p&M@h
w
% 2 0 (2*r^2 - 1) sqrt(3) `f(!i mN
% 2 2 r^2 * sin(2*theta) sqrt(6) @{bf]Oc
% 3 -3 r^3 * cos(3*theta) sqrt(8) E^rN)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) W uQdz&s>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) _*+M'3&=
% 3 3 r^3 * sin(3*theta) sqrt(8) Xd4~N:
% 4 -4 r^4 * cos(4*theta) sqrt(10) tlW}lN}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uJ%ql5XDV
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) E yNCky
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zy<0'k%U
% 4 4 r^4 * sin(4*theta) sqrt(10) __a9}m4i7x
% -------------------------------------------------- 3KqylC&.
% m~}nM |m%
% Example 1: GK)hK-
% hfY2pG9N
% % Display the Zernike function Z(n=5,m=1) ;;2s{{(R
% x = -1:0.01:1; AojX)_"z
% [X,Y] = meshgrid(x,x); p4/D%*G^`
% [theta,r] = cart2pol(X,Y); ]WS 7l@
% idx = r<=1; myPo&"_ x
% z = nan(size(X)); O)hNHIF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 6(eyUgnb
% figure 1PWDK1GI8
% pcolor(x,x,z), shading interp {3l]/X3
% axis square, colorbar 8garRB{
% title('Zernike function Z_5^1(r,\theta)') S -im
o
% gG#M-2P
% Example 2: DCHU=r
% \=w|Zeu{l
% % Display the first 10 Zernike functions V%"aU}
% x = -1:0.01:1; CrK}mbe
% [X,Y] = meshgrid(x,x); AH ;h#dT
% [theta,r] = cart2pol(X,Y); _- { > e
% idx = r<=1; 3t8VH`!mL{
% z = nan(size(X)); wz'D4B
% n = [0 1 1 2 2 2 3 3 3 3]; 1"i/*}M
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; .8k9yk
% Nplot = [4 10 12 16 18 20 22 24 26 28]; R@;kYS
% y = zernfun(n,m,r(idx),theta(idx)); d}Q;CF3m:
% figure('Units','normalized') C}7Sh6
% for k = 1:10 b8Y-!]F
% z(idx) = y(:,k); <_h
% subplot(4,7,Nplot(k)) SI-s:%O
% pcolor(x,x,z), shading interp yAaMYF@
% set(gca,'XTick',[],'YTick',[]) Mu TlN
% axis square "I
u3&mc
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1X]?-+',.
% end WxFVbtw
% [V
=O$X_
% See also ZERNPOL, ZERNFUN2. |'.\}xt7
G/b
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% Paul Fricker 11/13/2006 } DoNp[`
"1Vuf<?C
a8NL
% Check and prepare the inputs: )A,MTi
% ----------------------------- I t",WFE.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |
X! d*4
error('zernfun:NMvectors','N and M must be vectors.') :W^
k3/t
end qEE
V&
6,| !zaeS
if length(n)~=length(m) Z!DGCw
error('zernfun:NMlength','N and M must be the same length.') EP,lT.u3
end ;~F&b:CyG
!2=<MO
n = n(:); bDK72cQ
m = m(:); q9|'!m5K
if any(mod(n-m,2)) YB*I'm3q
error('zernfun:NMmultiplesof2', ... oUoDj'JN{
'All N and M must differ by multiples of 2 (including 0).') s>ilxLSX]
end JZB7?@h%
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if any(m>n) 4"7/+6Z
error('zernfun:MlessthanN', ... wX[g\,?}'
'Each M must be less than or equal to its corresponding N.') WTbq)D(&[_
end <<4U:
8(]*J8/wt
if any( r>1 | r<0 ) 22$M6Qof]n
error('zernfun:Rlessthan1','All R must be between 0 and 1.') p%[/
_ -7
end $9bLD
>.
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WS6'R
error('zernfun:RTHvector','R and THETA must be vectors.') NH~\kV
end muc6gwBp
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r = r(:); >cJf D9-<h
theta = theta(:); 6fY-DqF!
length_r = length(r); 0o7*5| T4
if length_r~=length(theta) c&X2k\
error('zernfun:RTHlength', ... ozB2L\D7
'The number of R- and THETA-values must be equal.') 8#L
V
oR
end Lh\ 1L
lub_2Cb|j
% Check normalization: m) QV2n
% -------------------- -?nr q <3
if nargin==5 && ischar(nflag) #p$iWY>e~
isnorm = strcmpi(nflag,'norm'); PUcxlD/a}
if ~isnorm 9?]69O
error('zernfun:normalization','Unrecognized normalization flag.') l$/.B=]
end y!eT>4Oyg
else A{|^_1
isnorm = false; 9lqH
end x18(}4
}l"pxp1K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \:y oS>G
% Compute the Zernike Polynomials %>Q[j`9y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Opavno%&
XCvL`
% Determine the required powers of r: 4]G J+a
% ----------------------------------- l$Y*ii
m_abs = abs(m); p?-qlPl
rpowers = []; _TntZv.?
for j = 1:length(n) zCji]:
rpowers = [rpowers m_abs(j):2:n(j)]; z|bAZKSRYx
end ;-kC&GZf
rpowers = unique(rpowers); O#MaZ.=
:_k5[KT.]9
% Pre-compute the values of r raised to the required powers, L0.F}~S
% and compile them in a matrix: qf
T71o(
% ----------------------------- *q; u%; 4
if rpowers(1)==0 -kzp>=
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); BD,J4xH;
rpowern = cat(2,rpowern{:}); <c3Te$.
rpowern = [ones(length_r,1) rpowern]; 7K5 tBUNQ
else U'@#n2p:k
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e1Q
rpowern = cat(2,rpowern{:}); C&HN#Q_
end F/
o }5H
3!M|Sf<s
% Compute the values of the polynomials: dOX"7kZ
% -------------------------------------- >npTUOGL=n
y = zeros(length_r,length(n)); [,L>5:T
for j = 1:length(n) >t#5eT`_ w
s = 0:(n(j)-m_abs(j))/2; fU<_bg
pows = n(j):-2:m_abs(j); G4rd<V0[D
for k = length(s):-1:1 S ^]mF>xX8
p = (1-2*mod(s(k),2))* ... (&MtK1;;
prod(2:(n(j)-s(k)))/ ... +I3j2u8L
prod(2:s(k))/ ... =&Z#QD"vl
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;F|8#! (
prod(2:((n(j)+m_abs(j))/2-s(k))); X'{o/U.
idx = (pows(k)==rpowers); nc3usq
y(:,j) = y(:,j) + p*rpowern(:,idx); "^Vnnb:Z*o
end I;Pd}A_}=_
jP#I](\eG
if isnorm t|P+^SL
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u-M Td
end NY?pvb
end 4s9qQ8?
% END: Compute the Zernike Polynomials GC`/\~TM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6<fcG
9zl-C*9vj
% Compute the Zernike functions: \
[bJ@f*."
% ------------------------------ L"RE[" m
idx_pos = m>0; 1}R\L"
idx_neg = m<0; 6zIK%<
V%'' GF
z = y; h<G7ocu !
if any(idx_pos) 9^7z"*@#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B[~Q0lPih
end G/ H>M%M
if any(idx_neg) 2y IDyo
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5,|of{8
end </pt($
iD.p KG
% EOF zernfun