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 x;aZ&
function z = zernfun(n,m,r,theta,nflag) e]fC!>w(\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Q:|E
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |(g2fByDf
% and angular frequency M, evaluated at positions (R,THETA) on the zwHsdB=v
% unit circle. N is a vector of positive integers (including 0), and y
+vcBuX
% M is a vector with the same number of elements as N. Each element UG$i5PV%i
% k of M must be a positive integer, with possible values M(k) = -N(k) ]F#kM21 1
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, T^>cT"ux_
% and THETA is a vector of angles. R and THETA must have the same >s~`K^zS
% length. The output Z is a matrix with one column for every (N,M) gE(03SX
% pair, and one row for every (R,THETA) pair. A
76yz`D
% 2ARh-zLb
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5?"ZM'4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z05pVe/5
% with delta(m,0) the Kronecker delta, is chosen so that the integral i:To8kdO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M-t9zT
% and theta=0 to theta=2*pi) is unity. For the non-normalized Jt][b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7.-|3Wcg
% 7T78S&g
% The Zernike functions are an orthogonal basis on the unit circle. JH9CN
% They are used in disciplines such as astronomy, optics, and tO$M[P=b
% optometry to describe functions on a circular domain. !;oBvE7Kh
% 2x
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% The following table lists the first 15 Zernike functions. 1' s^W
% Ado>)c"*y1
% n m Zernike function Normalization 5#tvc4+)
% -------------------------------------------------- xRmB?kM3]5
% 0 0 1 1 )VrHP9fu
% 1 1 r * cos(theta) 2 u]-$]zIH
% 1 -1 r * sin(theta) 2 :PJjy6,1
% 2 -2 r^2 * cos(2*theta) sqrt(6) )JON&~C
% 2 0 (2*r^2 - 1) sqrt(3) nMqU6X>P!
% 2 2 r^2 * sin(2*theta) sqrt(6) 'UCL?$
% 3 -3 r^3 * cos(3*theta) sqrt(8) >~k
Y{_
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0jMrL\>C
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b9Nw98`
% 3 3 r^3 * sin(3*theta) sqrt(8) c$TBHK;c
% 4 -4 r^4 * cos(4*theta) sqrt(10) -#h
\8Xl
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kS>j!U(%d
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) A,@"(3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &3MHe$
% 4 4 r^4 * sin(4*theta) sqrt(10) j\<S 6%p#R
% -------------------------------------------------- 54-x 14")
% I;LqyzM
% Example 1: na>B{6
% 7UfyOOFa
% % Display the Zernike function Z(n=5,m=1)
&0myA_So
% x = -1:0.01:1; 5NK:94&JE
% [X,Y] = meshgrid(x,x); =Vfj#WL
% [theta,r] = cart2pol(X,Y); J2-xnUa]7
% idx = r<=1; F);C?SW"
% z = nan(size(X)); ^;e`ZtcI
% z(idx) = zernfun(5,1,r(idx),theta(idx)); mjpH)6aD0
% figure Vj1AW<
% pcolor(x,x,z), shading interp Z2r\aZ-d`
% axis square, colorbar .x&>H
% title('Zernike function Z_5^1(r,\theta)') gKnAw+u\
% Iq9+
% Example 2: v%r! }s
% m`|+_{4[n
% % Display the first 10 Zernike functions /TdTo@
% x = -1:0.01:1; WO^h\#^n
% [X,Y] = meshgrid(x,x); 6+>rf{5P7
% [theta,r] = cart2pol(X,Y); f>o@Y]/l
% idx = r<=1;
FM5$83Q
% z = nan(size(X)); Sq,x@
% n = [0 1 1 2 2 2 3 3 3 3]; $%<gp@Gz
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; M&L" yQA
% Nplot = [4 10 12 16 18 20 22 24 26 28]; BdSTB"
% y = zernfun(n,m,r(idx),theta(idx)); 4)?c[aC4P
% figure('Units','normalized') X~0P+E#
% for k = 1:10 Wr;)3K
% z(idx) = y(:,k); yq2Bz7P
% subplot(4,7,Nplot(k)) B}p/ ,4x6
% pcolor(x,x,z), shading interp wI:oe`?H
% set(gca,'XTick',[],'YTick',[]) ie)Qsw@
% axis square H74hv`G9
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) MFVFr "
% end {.ph)8
% /dO&r'!:
% See also ZERNPOL, ZERNFUN2. ~0`Pe{^*
WH!<Z=#c}
% Paul Fricker 11/13/2006 @Q'5/q+
3|C"F-'<
IQ\`n|
% Check and prepare the inputs: >DDQ7
l
% ----------------------------- j \SDw
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yy9Bd>
error('zernfun:NMvectors','N and M must be vectors.') u%2u%-w
end v / a/
]uP{Sj
if length(n)~=length(m) McfSB(59
error('zernfun:NMlength','N and M must be the same length.') U+W8)7bc
end #ws6z`mt
.UJk0%1
n = n(:); r J&1[=s
m = m(:); Wd[XQZ<
if any(mod(n-m,2)) >k:)'*
error('zernfun:NMmultiplesof2', ... q,2
@X~T
'All N and M must differ by multiples of 2 (including 0).') Cnc77EUD
end z*FlZLHY
bw@tA7Y
if any(m>n) ?p`}6s Q}
error('zernfun:MlessthanN', ... ?Hy++
'Each M must be less than or equal to its corresponding N.') d(k`Yk8
end yfV{2[8ux
3p7*UVR"
if any( r>1 | r<0 ) 3#dUQ1qo6
error('zernfun:Rlessthan1','All R must be between 0 and 1.') :yv!
x
end \4V'NTjB
9t= erhUr
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J9/w_,,R$
error('zernfun:RTHvector','R and THETA must be vectors.') NvYgRf}uh
end }D0j%~&"e
%e_WO,R
r = r(:); !\p-|51
theta = theta(:); 8z@A/$T
length_r = length(r); e{"d6pF=
if length_r~=length(theta) 6~^+</?
error('zernfun:RTHlength', ... Yd]f}5F
'The number of R- and THETA-values must be equal.') L&l>?"_
end lVMAab
B^eea [
% Check normalization: Q&wBX%@^L
% -------------------- JG4Tb{F=
if nargin==5 && ischar(nflag) |s|RJA1
isnorm = strcmpi(nflag,'norm'); j+ s8V-7(
if ~isnorm K":-zS
error('zernfun:normalization','Unrecognized normalization flag.') 2 !{P<
end zm"& 8/l
else N#|c2n+
isnorm = false; IN_GL18^MV
end 1`b?nX
wp$SO^?-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u K 8r
% Compute the Zernike Polynomials ^ 3Vjmv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NmK%k jCx
N$p O] p
% Determine the required powers of r: 6Bs_"
P[
% ----------------------------------- WpRi+NC}ln
m_abs = abs(m); KPKby?qQ^
rpowers = []; !iITX,'8
for j = 1:length(n) UGl}=hwKkG
rpowers = [rpowers m_abs(j):2:n(j)]; )-[X^l
j
end Jg^tr>I~
rpowers = unique(rpowers); 8iq~ha$]|
S&@~F|
% Pre-compute the values of r raised to the required powers, OG0ro(|dI
% and compile them in a matrix: ^fH]Rlx
% ----------------------------- (gz|6N
if rpowers(1)==0 *_U
z**M
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _M{m6k(h
rpowern = cat(2,rpowern{:}); a ipvG
rpowern = [ones(length_r,1) rpowern]; 2Ask]
else k5W5 9tz
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); eDG=-a4
rpowern = cat(2,rpowern{:}); tWD*uAb
end yv,9 0+k
))u$j4V
% Compute the values of the polynomials: }i?P(
Au
% -------------------------------------- 2uV=kq nO
y = zeros(length_r,length(n)); cND2(<jx:
for j = 1:length(n) HnZrRHT0
s = 0:(n(j)-m_abs(j))/2; nbhx2@Teqe
pows = n(j):-2:m_abs(j); Dr<% Lr
for k = length(s):-1:1 ;p1%KmK3
p = (1-2*mod(s(k),2))* ... Nqz-Mr`
prod(2:(n(j)-s(k)))/ ... !dGy"-i$h
prod(2:s(k))/ ... ">NBPanJ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H`m|R
prod(2:((n(j)+m_abs(j))/2-s(k))); .b5B7x}
idx = (pows(k)==rpowers); 8ec~"vGLz~
y(:,j) = y(:,j) + p*rpowern(:,idx); L<J%IlcfO
end t:$p8qR
v='7.A
if isnorm @^/JNtbH!
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yP~D."
end dEns|r
end <"aPoGda
% END: Compute the Zernike Polynomials sg{>-KHM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fpl<2eBg4
SbrBlP:G
% Compute the Zernike functions: j =[Td
% ------------------------------ 4LKOBiEM
idx_pos = m>0; RVX-3FvP
idx_neg = m<0; dAohj
QH:
N!^U{;X7/
z = y; .#EmE'IP*
if any(idx_pos) ln#Lx&r;|
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !)KX?i[Q
end ?zKDPBj
if any(idx_neg) ^BSMlKyB
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); L@9"6&
end Mt<TEr}7Z=
B4_0+K H
% EOF zernfun