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 ,K
8R%B
function z = zernfun(n,m,r,theta,nflag) "m4._4U
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. TkBHlTa"=
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Py y!B
% and angular frequency M, evaluated at positions (R,THETA) on the nm Y_ )s
% unit circle. N is a vector of positive integers (including 0), and C3)*Mn3%P
% M is a vector with the same number of elements as N. Each element .o8Sy2PaV
% k of M must be a positive integer, with possible values M(k) = -N(k) JuQwZ]3ed
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]l>LU2 sx
% and THETA is a vector of angles. R and THETA must have the same WPI<SsLd
% length. The output Z is a matrix with one column for every (N,M) /W9(}Id6
% pair, and one row for every (R,THETA) pair. {7'Wi$^F
% ;x%"o[[>
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /#jH#f[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 0$JH5RC
% with delta(m,0) the Kronecker delta, is chosen so that the integral `,QcOkvbC
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KW-GVe%8f
% and theta=0 to theta=2*pi) is unity. For the non-normalized |W_;L6)
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2,aH1Xbex
% o=J-Ju
% The Zernike functions are an orthogonal basis on the unit circle. ~I6N6T Z
% They are used in disciplines such as astronomy, optics, and lg"aB
% optometry to describe functions on a circular domain. _Ne fzZWUJ
% !6!Gx:
% The following table lists the first 15 Zernike functions. )G#mC0?PV
% =' uePM")
% n m Zernike function Normalization *:bexD H
% -------------------------------------------------- bd]9kRq1K
% 0 0 1 1 0vX4v)-^u
% 1 1 r * cos(theta) 2 >3ax `8
% 1 -1 r * sin(theta) 2 Xii>?sA5Z"
% 2 -2 r^2 * cos(2*theta) sqrt(6) "i#aII+T
% 2 0 (2*r^2 - 1) sqrt(3) 0civXZgj
% 2 2 r^2 * sin(2*theta) sqrt(6) \?Sv O
% 3 -3 r^3 * cos(3*theta) sqrt(8) <qg4Rz\c]
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ijsoY\V50
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) $Nd,6w*`
% 3 3 r^3 * sin(3*theta) sqrt(8) (\0
<|pW
% 4 -4 r^4 * cos(4*theta) sqrt(10) rk6K0TQ8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I4W@t4bZ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8~tX>q<@q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2n)?)w]!M
% 4 4 r^4 * sin(4*theta) sqrt(10) KL3Z(
% -------------------------------------------------- h PL]B_<
%
C];P yQS
% Example 1: v3#,Z!
% oNZ_7tU
% % Display the Zernike function Z(n=5,m=1) 0:f]&Ng
% x = -1:0.01:1; \?pyax8
% [X,Y] = meshgrid(x,x); Y{D%v
% [theta,r] = cart2pol(X,Y); 8[;vC$
% idx = r<=1; _0(%^5Y
% z = nan(size(X)); S=(<m%f
% z(idx) = zernfun(5,1,r(idx),theta(idx)); k,[*h-{8
% figure jUE gu
% pcolor(x,x,z), shading interp s3HVX'
% axis square, colorbar Jy5sZ}t[
% title('Zernike function Z_5^1(r,\theta)') baBBn%_V
% B*N1)J\5
% Example 2: jMgXIK\
% Hs*["zFc
% % Display the first 10 Zernike functions ,Cb3R|L8
% x = -1:0.01:1; #8|LPfA
% [X,Y] = meshgrid(x,x); ?u|@,tQ[
% [theta,r] = cart2pol(X,Y); ]I [~0PCSX
% idx = r<=1; z%OKv[/N
% z = nan(size(X)); )rq |t9kix
% n = [0 1 1 2 2 2 3 3 3 3]; -
8p!,+Dk
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; PD)"od
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 7~SwNt,
% y = zernfun(n,m,r(idx),theta(idx)); x2rAB5r6
% figure('Units','normalized') l-Z( ]
% for k = 1:10 I |U'@E
% z(idx) = y(:,k); p&h?p\IF
% subplot(4,7,Nplot(k)) {uj_4Ft
% pcolor(x,x,z), shading interp lj (y
% set(gca,'XTick',[],'YTick',[]) .qgUD
% axis square X_]rtG
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LWyr
% end N%
4"9K
% 'v"=
% See also ZERNPOL, ZERNFUN2. X` zWw_i
<7M-?g:vj
% Paul Fricker 11/13/2006 8NWo)y49H
r- <O'^C
G3~`]qf
% Check and prepare the inputs: TaN{xpo
% ----------------------------- gcU*rml
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;f[lq^eV
error('zernfun:NMvectors','N and M must be vectors.') :OG I|[
end c-sjYJXKM*
U[@y8yN6M
if length(n)~=length(m) 5o#JHD
error('zernfun:NMlength','N and M must be the same length.') >2'"}np*
end zaqX};b
Cf
2@x
n = n(:); cJ;Nh>ey
m = m(:); wI$a1H
if any(mod(n-m,2)) wDJ`#"5p{
error('zernfun:NMmultiplesof2', ... n t}7|h|
'All N and M must differ by multiples of 2 (including 0).') =]Vz=<
end Xw-[Sf]p
Ao\xse{E
if any(m>n) c.ow4~>
error('zernfun:MlessthanN', ... Yc:%2KZ"
'Each M must be less than or equal to its corresponding N.') |eqBCZn
end |'Fe?~P`
V'Gal`
if any( r>1 | r<0 ) R4m{D
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0!T`.UMI
end @^P^-B
OT9]{|7
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $ik*!om5
error('zernfun:RTHvector','R and THETA must be vectors.') 7>FXsUt_
end p+VU:%.t
9iA rBL"
r = r(:); :DD<0
theta = theta(:); )cqD"> vs
length_r = length(r); l8\UO<^fY
if length_r~=length(theta) Ri. tA
error('zernfun:RTHlength', ... Zh"m;l/]
'The number of R- and THETA-values must be equal.') >f(?Mxh2
end "Ms;sdjg}&
?=VvFfv%
% Check normalization: T5S4,.o9W
% -------------------- >STtX6h
if nargin==5 && ischar(nflag) J|`0GDSn
isnorm = strcmpi(nflag,'norm'); +yGQt3U
if ~isnorm rE3dHJN;
error('zernfun:normalization','Unrecognized normalization flag.') *g/klK
end XLNbV?
else ag-A}k>v
isnorm = false; =>jp\A
end ekM?
'9ez
Cp8=8N(Xb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [q<'ty
% Compute the Zernike Polynomials E+f)Zg
:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XYEwn_Y
^L[:DB{Z
% Determine the required powers of r: U!wi;W2
% ----------------------------------- dbI>\khI
m_abs = abs(m); OQVrg2A%(
rpowers = []; bsIG1&n'T
for j = 1:length(n) zWHq4@K
rpowers = [rpowers m_abs(j):2:n(j)]; R><g\{G]
end wQ}r/2n|^
rpowers = unique(rpowers); Z_d"<k}I
h9vcN#22D
% Pre-compute the values of r raised to the required powers, i5,iJe0cA
% and compile them in a matrix: NGx3f3 9
% ----------------------------- %opBJ
if rpowers(1)==0 }3pM,.
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Q;M\fBQO}&
rpowern = cat(2,rpowern{:}); i"8mrWb
rpowern = [ones(length_r,1) rpowern]; T]#V
else :^;c(>u{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }z3j7I
rpowern = cat(2,rpowern{:}); =2Y;)wrF
end jr6_|(0
i6
K1&
QAXyP
% Compute the values of the polynomials: 'h>uR|
% -------------------------------------- x7j#@C
y = zeros(length_r,length(n)); _(W@FS
for j = 1:length(n) &#r+a'
s = 0:(n(j)-m_abs(j))/2; 8{ zX=
pows = n(j):-2:m_abs(j); 6{Wo5O{!\
for k = length(s):-1:1 -YRIe<}E -
p = (1-2*mod(s(k),2))* ... I>c,Bo7
prod(2:(n(j)-s(k)))/ ... u-_r2U
prod(2:s(k))/ ... s#2t\}/
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... fgLjF,Y
prod(2:((n(j)+m_abs(j))/2-s(k))); dzVi ~wt_&
idx = (pows(k)==rpowers); g=*jKSZ
y(:,j) = y(:,j) + p*rpowern(:,idx); Zk3Pv0c
end m[hL
GD'Fi
IqOg{#sm
if isnorm 2
$>DX\h
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 12$0-@U
end 8@3K, [Mo
end QY\k3hiqn
% END: Compute the Zernike Polynomials JA^o/%a^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rK3kg2H
PEMkx"h +
% Compute the Zernike functions: rp Nb.
% ------------------------------ 6j#JhcS+
idx_pos = m>0; ,75)
idx_neg = m<0; KA3U W
\pmS*Dt
z = y; qi-XNB`b
if any(idx_pos) m[DQ;`Y
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _Q V=3UWP
end +WX/4_STV
if any(idx_neg) `lf_wB+I
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); kA:Y^2X'
end SzULy
>e
AGBV7Kk
% EOF zernfun