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 j,Y=GjfGM
function z = zernfun(n,m,r,theta,nflag) VCI G+Gz
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Q_Rr5/
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N oKUJB.PF
% and angular frequency M, evaluated at positions (R,THETA) on the 01J.XfCd6
% unit circle. N is a vector of positive integers (including 0), and d 9|u~3
% M is a vector with the same number of elements as N. Each element /T?['#:r-)
% k of M must be a positive integer, with possible values M(k) = -N(k) )9$Xfq/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, a)]N#gx
% and THETA is a vector of angles. R and THETA must have the same *m2:iChY
% length. The output Z is a matrix with one column for every (N,M) KM6r}CDHs
% pair, and one row for every (R,THETA) pair. jm!G@k6TA
% <H.Ml>q:r
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j JW0a\0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j$,`EBf`:<
% with delta(m,0) the Kronecker delta, is chosen so that the integral 8p5u1 ;2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, IzG7!K
% and theta=0 to theta=2*pi) is unity. For the non-normalized Ky+TgR
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,,?t>|3
% vR -/c
% The Zernike functions are an orthogonal basis on the unit circle. $ysC)5q.
% They are used in disciplines such as astronomy, optics, and c7'Pzb)'
% optometry to describe functions on a circular domain. .gB#g{5+J
% (g 8K?Q
% The following table lists the first 15 Zernike functions. [mhY_Hmz]
% !!9V0[
% n m Zernike function Normalization x`$4
% -------------------------------------------------- E0YXgQa
% 0 0 1 1 M/BBNT
% 1 1 r * cos(theta) 2 9s}--_k?F2
% 1 -1 r * sin(theta) 2 DpA)Z??
% 2 -2 r^2 * cos(2*theta) sqrt(6) ^OUkFH;dG?
% 2 0 (2*r^2 - 1) sqrt(3) |XQ!xFB
% 2 2 r^2 * sin(2*theta) sqrt(6) `.n[G~*w~1
% 3 -3 r^3 * cos(3*theta) sqrt(8) aw(P@9]
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ^ H'|iju
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) GDk/85cv0$
% 3 3 r^3 * sin(3*theta) sqrt(8) lGxG$0`;;
% 4 -4 r^4 * cos(4*theta) sqrt(10) s3q65%D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VBOq~>V6(v
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) L%!jj7,9-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sYvO"|
% 4 4 r^4 * sin(4*theta) sqrt(10) h4V.$e<T&
% -------------------------------------------------- x.'O_7c0:
% DJeG
% Example 1: EPyFM_k
% UlyX$f%2
% % Display the Zernike function Z(n=5,m=1) f F?=W
% x = -1:0.01:1; k+&| *!j
% [X,Y] = meshgrid(x,x); JTVCaL3Z
% [theta,r] = cart2pol(X,Y); !x>P]j7A}Y
% idx = r<=1; MLUq"f~ N
% z = nan(size(X)); t.NG]ejZ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); BONM:(1
% figure *NDzU%X8
% pcolor(x,x,z), shading interp pCv=rK@
% axis square, colorbar $AoN,B>
% title('Zernike function Z_5^1(r,\theta)') k*M1m'1
% gCd9"n-e
% Example 2: i2EB.Zlv
% #\w~(Nm-
% % Display the first 10 Zernike functions # *\PU
% x = -1:0.01:1; HdVGkv/
% [X,Y] = meshgrid(x,x); *K!V$8k=99
% [theta,r] = cart2pol(X,Y); ,rQznE1e
% idx = r<=1; zL1H[}[z+
% z = nan(size(X)); _uL m !ku
% n = [0 1 1 2 2 2 3 3 3 3]; !
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; I( pU_7mw
% Nplot = [4 10 12 16 18 20 22 24 26 28]; X)`?P*[
% y = zernfun(n,m,r(idx),theta(idx)); R(3V !ph
% figure('Units','normalized') SZEX;M
% for k = 1:10 a Z
^SK|E
% z(idx) = y(:,k); JIDE]f
% subplot(4,7,Nplot(k)) Yk[yG;W
% pcolor(x,x,z), shading interp ]ZZ7j
% set(gca,'XTick',[],'YTick',[]) !qT.D:!@zF
% axis square Aqq%HgY:t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #AE'arT<
% end ]x! vPIyq
% amOBUD5Ld`
% See also ZERNPOL, ZERNFUN2. "h\{PoG
^BW V6
% Paul Fricker 11/13/2006 ]e 81O#t3
Bx2E9/S3
}wz )"
% Check and prepare the inputs: u.R:/H<>~
% ----------------------------- J=5G<
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tvZpm@1
error('zernfun:NMvectors','N and M must be vectors.') g;Bq#/w
end BHqJ~2&FDW
H"6:!;9,
if length(n)~=length(m) ewD61Y8-
error('zernfun:NMlength','N and M must be the same length.') + ,0RrD )
end 7'd_]e-.
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n = n(:); IKH#[jW'IB
m = m(:); }>fL{};Z"
if any(mod(n-m,2)) |{<g-)
error('zernfun:NMmultiplesof2', ... *[k7KG2_U
'All N and M must differ by multiples of 2 (including 0).') J8~3LE
)G
end YB.r-c"Y
lhKd<Y"
if any(m>n) >DpnIWn
error('zernfun:MlessthanN', ... e=QnGT*b5
'Each M must be less than or equal to its corresponding N.') UII R$,XB
end oe# :EfT
Fn yA;,*
if any( r>1 | r<0 ) %
=br-c
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .^fq$7Y}7
end 77.5
_
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #Q"el3P+q
error('zernfun:RTHvector','R and THETA must be vectors.') 5,|^4
ZA
end /!ux P~2U
lmgMR|v
r = r(:); _\1wLcFj
theta = theta(:); JIQS'r
length_r = length(r); ;N6L`|
if length_r~=length(theta) zH.DyD5T;
error('zernfun:RTHlength', ... |r$Vb$z
'The number of R- and THETA-values must be equal.') -6aGcPq
end 8J7xs6@
9Ld3
% Check normalization: &Dgho
% -------------------- "n=`{~F
if nargin==5 && ischar(nflag) Da0E)
isnorm = strcmpi(nflag,'norm'); ]+{Cy\*kR
if ~isnorm H_3S#.
error('zernfun:normalization','Unrecognized normalization flag.') 1BmevEa)
end {;=I69X
else AM#VRRTU
isnorm = false; dyC: Mko=
end l%oie1g l
dc|"34;^"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b,cA mZ
% Compute the Zernike Polynomials ;lB%N
t<,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b`usRoD{+
SL?
!
RQ
% Determine the required powers of r:
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% ----------------------------------- 1>[3(o3t
m_abs = abs(m); m1heU3BUWU
rpowers = []; kS%FV;9>(
for j = 1:length(n)
lc,{0$
1<
rpowers = [rpowers m_abs(j):2:n(j)]; DvKM[z3j
end ;oH17
rpowers = unique(rpowers); HpC|dtro
By]XD~gcP
% Pre-compute the values of r raised to the required powers, U@@#f;&
% and compile them in a matrix: s7A{<>:
% ----------------------------- ce0TQ
if rpowers(1)==0 MS)# S&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h/?8F^C#v
rpowern = cat(2,rpowern{:}); bN`oQ.Z 4
rpowern = [ones(length_r,1) rpowern]; S#8wnHq
else :Ag]^ot
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); f<=
#WV
rpowern = cat(2,rpowern{:}); O5CIK}A
end 2l}FOdq
(`5No:?v<
% Compute the values of the polynomials: Oz#$x
% -------------------------------------- w}c1zpa
y = zeros(length_r,length(n)); M}k )Ep9
for j = 1:length(n) r :{2}nE
s = 0:(n(j)-m_abs(j))/2; e#(0af8A
pows = n(j):-2:m_abs(j); #UG| \}Lp
for k = length(s):-1:1 /pan{.< k
p = (1-2*mod(s(k),2))* ... E{[c8l2B
prod(2:(n(j)-s(k)))/ ... zW,m3~XX:
prod(2:s(k))/ ... T;XEU%:LK
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... bHH{bv~Z
prod(2:((n(j)+m_abs(j))/2-s(k))); CkE@Ll3Z
idx = (pows(k)==rpowers); TG8QT\0G
y(:,j) = y(:,j) + p*rpowern(:,idx); ^0_ *AwIcN
end <W2}^q7F^
iA3d[%tBb
if isnorm {a.{x+!5I-
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~
'
81
end _A|1_^[G(
end yH#zyO4fD-
% END: Compute the Zernike Polynomials `<i|K*u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q6@}t&k4C
=uQCm#
% Compute the Zernike functions: UK*+EEv
% ------------------------------ sesr`,m.,
idx_pos = m>0; M7-piRnd4
idx_neg = m<0; 0APwk
}
\:mx Ri
z = y; VI,z7
\
if any(idx_pos) Z#BwJHh
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %H75u6
end B(wk $2
if any(idx_neg) kbJ/7
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); C(Ujx=G+3
end @+h2R
QDYS}{A:V
% EOF zernfun