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 aC`Li^
function z = zernfun(n,m,r,theta,nflag) Bb~5& @M|N
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. :3v9h^|+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8=:A/47=J
% and angular frequency M, evaluated at positions (R,THETA) on the wTTRoeJ}
% unit circle. N is a vector of positive integers (including 0), and L^lS^P
% M is a vector with the same number of elements as N. Each element &`\ ep9
% k of M must be a positive integer, with possible values M(k) = -N(k) [q'eENG
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @8|Gh]\P
% and THETA is a vector of angles. R and THETA must have the same _ j~4+H
% length. The output Z is a matrix with one column for every (N,M) $57\u/(
% pair, and one row for every (R,THETA) pair. 3c b[RQf
% B22b&0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m$?.Yig?
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H"_v+N5=
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;d4y{
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d<#p %$A4
% and theta=0 to theta=2*pi) is unity. For the non-normalized D3y>iQd
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TFO74^
% 3Y`>6A=
% The Zernike functions are an orthogonal basis on the unit circle. ZW>o5x__b
% They are used in disciplines such as astronomy, optics, and |) O):
% optometry to describe functions on a circular domain. H<,bq*@
% #pX8{Tf[
% The following table lists the first 15 Zernike functions. $.a|ae|K
% >PIPp7C
% n m Zernike function Normalization Xtkw Z3
% -------------------------------------------------- u#FXW_-TK
% 0 0 1 1 &3I$8v|!?
% 1 1 r * cos(theta) 2 /_q#ah
% 1 -1 r * sin(theta) 2 BhLZ7 *
% 2 -2 r^2 * cos(2*theta) sqrt(6) gGI8t@t:
% 2 0 (2*r^2 - 1) sqrt(3) (etUEb^}T
% 2 2 r^2 * sin(2*theta) sqrt(6) `y2ljIWJ
% 3 -3 r^3 * cos(3*theta) sqrt(8) eES'}[W>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) wlr Ign%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *Rq`*D>:U}
% 3 3 r^3 * sin(3*theta) sqrt(8) ^7Lk-a7gp
% 4 -4 r^4 * cos(4*theta) sqrt(10) #&V5H{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y''6NGf
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2 5Q+1
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /ERNS/w
% 4 4 r^4 * sin(4*theta) sqrt(10) "R23Pi
% -------------------------------------------------- @0|nq9l1
% ")ED)&e
% Example 1: uf]Y^,2
% Rboof`pVt
% % Display the Zernike function Z(n=5,m=1) @^!\d#/M
% x = -1:0.01:1; Ukc'?p,*
% [X,Y] = meshgrid(x,x); f*<ps
o
% [theta,r] = cart2pol(X,Y); =&2$/YX0D
% idx = r<=1; -2 xE#r
% z = nan(size(X)); y\#o2PVmY
% z(idx) = zernfun(5,1,r(idx),theta(idx)); s`c?:
% figure x%6hM|U
% pcolor(x,x,z), shading interp c4 5?St
% axis square, colorbar H* /&A9("
% title('Zernike function Z_5^1(r,\theta)') 4gOgWBv
% :G 5C ]'t
% Example 2: 1~@|eWr|
% Szts<n5
% % Display the first 10 Zernike functions %K zbO0
% x = -1:0.01:1; _R74/|
% [X,Y] = meshgrid(x,x); vLDi ;
% [theta,r] = cart2pol(X,Y); !BUi)mo
% idx = r<=1; t8vc@of$c,
% z = nan(size(X)); TEWAZVE*
% n = [0 1 1 2 2 2 3 3 3 3]; mgVML&^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; bMmra.x4L
% Nplot = [4 10 12 16 18 20 22 24 26 28]; uNbIX:L,
% y = zernfun(n,m,r(idx),theta(idx)); &SmXI5>Bo0
% figure('Units','normalized') EwQae(PpA
% for k = 1:10 .&iN(Bd
% z(idx) = y(:,k); ltSh'w0
% subplot(4,7,Nplot(k)) y]'CXCml)
% pcolor(x,x,z), shading interp p=B?/Sqa
% set(gca,'XTick',[],'YTick',[]) -k{Jp/-D
% axis square @9vvR7{P
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) oLS7`+b$
% end !M(:U,?B
% r6t&E%b
% See also ZERNPOL, ZERNFUN2. ~ziexZ=N
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% Paul Fricker 11/13/2006 )6{P8k4Zr
GV8)Kor%
%[Zz0|A
% Check and prepare the inputs: S}cF0B1E*
% ----------------------------- s.:r;%a
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s;1e0n
error('zernfun:NMvectors','N and M must be vectors.') cPuHLwwYf
end _{Y$o'*#I
_~A~+S}
if length(n)~=length(m) 9m8ee&,
error('zernfun:NMlength','N and M must be the same length.') ? )_7U
end 0d4cE10
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n = n(:); I4ilR$jg
m = m(:); h8=h >W-
if any(mod(n-m,2)) D|Si)_
Iz
error('zernfun:NMmultiplesof2', ... zfjw;sUX
'All N and M must differ by multiples of 2 (including 0).') Rp/-Pv
end T~J?AKx
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if any(m>n) &fSTR-8ev#
error('zernfun:MlessthanN', ... J+Bdz6lt
'Each M must be less than or equal to its corresponding N.') e{C6by"j{S
end "'A"U
_tj&Psp
if any( r>1 | r<0 ) r)b<{u=]
error('zernfun:Rlessthan1','All R must be between 0 and 1.') !8$RBD %
end qks|d_
O
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yd>}wHt
error('zernfun:RTHvector','R and THETA must be vectors.') Of`c`-<j
end kon=il<@
'ere!:GJD
r = r(:); 1TRN~#ix
theta = theta(:); lLCdmxbT
length_r = length(r); / Z!i;@Wf
if length_r~=length(theta) \ e,?rH
error('zernfun:RTHlength', ... g$3>~D
'The number of R- and THETA-values must be equal.') @ Nb%L&=P8
end IKcKRw/O$
a+?~;.i~
% Check normalization: %MJ;Q?KB
% -------------------- HarFE4V
if nargin==5 && ischar(nflag) 1q]c7"
isnorm = strcmpi(nflag,'norm'); !Iq{ 5:
if ~isnorm \L[i9m| e
error('zernfun:normalization','Unrecognized normalization flag.') H06Bj(Y!
end CLN+I'uX0
else Nn#u%xvJt
isnorm = false; 6vp0*ww
end NHiq^ojk
=Od>;|]m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dg2uE8k
% Compute the Zernike Polynomials FC}oL"kk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n}J^6:1
s#^pC*,'
% Determine the required powers of r: 1r571B*O
% ----------------------------------- +v15[^F
m_abs = abs(m); >V!LitdJ
rpowers = []; j>'B[
for j = 1:length(n) _N'75
rpowers = [rpowers m_abs(j):2:n(j)]; arh@`'Q
end qY# d+F,t
rpowers = unique(rpowers); jJ++h1
K
`="v>qN2\
% Pre-compute the values of r raised to the required powers, aqr!oxn?t
% and compile them in a matrix: ;V.vfar
% ----------------------------- J_xG}d
if rpowers(1)==0 -7`-wu
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7X'y>\^w^>
rpowern = cat(2,rpowern{:}); _Bk
U+=|J
rpowern = [ones(length_r,1) rpowern]; b3U6;]|x
else *gu8-7'
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'IQsve7cI
rpowern = cat(2,rpowern{:}); HDS"F.l5
end SRz&Nb
R^P_{_I*"
% Compute the values of the polynomials: ?~F. /
% -------------------------------------- /EFq#+6
y = zeros(length_r,length(n)); :oa9#c`L
for j = 1:length(n) $TG?4
s = 0:(n(j)-m_abs(j))/2; $a.u05
pows = n(j):-2:m_abs(j); /f3m)pT
for k = length(s):-1:1 G)7)]yBL
p = (1-2*mod(s(k),2))* ... =!<G!^
prod(2:(n(j)-s(k)))/ ... >oqZ !V5[
prod(2:s(k))/ ... OE"<!oIs
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... E
$6ejGw-
prod(2:((n(j)+m_abs(j))/2-s(k))); DQgH_!
idx = (pows(k)==rpowers); ybvI?#
y(:,j) = y(:,j) + p*rpowern(:,idx); I@./${o
end Y60"M4j
+1@AGJU3
if isnorm Q4K+*Fi}
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |:2c$zq
end C\Ayv)S#2
end Hj~O49%j&
% END: Compute the Zernike Polynomials Lq04T0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q}P-$X+/ n
E`AYee%l
% Compute the Zernike functions: g6euXI
% ------------------------------ $D_HZ"ytu
idx_pos = m>0; }lfn0 %(@
idx_neg = m<0; 0IzZKRw
:p-Y7CSSu
z = y; xo~g78jm7,
if any(idx_pos) u!1/B4!'O
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); T[2}p=<%
end 4/MNqit+
if any(idx_neg) #s+Q{2s
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tWk{1IL
end !F7: i
`K?1L{p'4
% EOF zernfun