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 VGfMN|h
function z = zernfun(n,m,r,theta,nflag) tkVbo.[8K
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. NS9B[*"Jl
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S\''e`Eb"5
% and angular frequency M, evaluated at positions (R,THETA) on the l]@&D#3ZM
% unit circle. N is a vector of positive integers (including 0), and }XZ'v_Ti
% M is a vector with the same number of elements as N. Each element I[=j&rK`
% k of M must be a positive integer, with possible values M(k) = -N(k) 2{]`W57_=
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, V_>\9m
% and THETA is a vector of angles. R and THETA must have the same pwO>h>ik
% length. The output Z is a matrix with one column for every (N,M) G3{Q"^S"
% pair, and one row for every (R,THETA) pair. M^MdRu
% TK5K_V*7
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike il}%7b-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4,..kSA3iw
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?f#y1m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s4G|_==
% and theta=0 to theta=2*pi) is unity. For the non-normalized T#M,~lD
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O>qll6]{@
% R#xCkl -
% The Zernike functions are an orthogonal basis on the unit circle. v$~QU{&
% They are used in disciplines such as astronomy, optics, and ]Gpxhg
% optometry to describe functions on a circular domain. V7GRA#|
% @_U;9)
% The following table lists the first 15 Zernike functions. @'YS1 N<
% ~;O v-^tp
% n m Zernike function Normalization @*}D$}aR'V
% -------------------------------------------------- A&s:\3*Kh
% 0 0 1 1 k xP-,MD
% 1 1 r * cos(theta) 2 HqI t74+
% 1 -1 r * sin(theta) 2 EM]s/LD@%
% 2 -2 r^2 * cos(2*theta) sqrt(6) O>SLOWgha
% 2 0 (2*r^2 - 1) sqrt(3) (2$(
?-M
% 2 2 r^2 * sin(2*theta) sqrt(6) lFa02p0
% 3 -3 r^3 * cos(3*theta) sqrt(8) e@c0WlWa
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Kpb#K[(]&
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 4?0vso*X<:
% 3 3 r^3 * sin(3*theta) sqrt(8) E8>Rui@9
% 4 -4 r^4 * cos(4*theta) sqrt(10) h lkn%
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .nG#co"r}3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) q+P|l5_
t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8S>&WR%jH]
% 4 4 r^4 * sin(4*theta) sqrt(10) 'I_Qb$
% -------------------------------------------------- F_Z- 8>P
% 9U{a{~b
% Example 1: K|Ld,bq
% #6ri-n
% % Display the Zernike function Z(n=5,m=1) 5:O-tgig.
% x = -1:0.01:1; ;w:M`#2
% [X,Y] = meshgrid(x,x); MG[o%I96
% [theta,r] = cart2pol(X,Y); ;epV<{e$q4
% idx = r<=1; 8dV=[+
% z = nan(size(X)); 7#@cz5Su
% z(idx) = zernfun(5,1,r(idx),theta(idx)); W4[V}s5u
% figure ~vs}.kb
% pcolor(x,x,z), shading interp 5Ycco,x
% axis square, colorbar u1t%(_h
% title('Zernike function Z_5^1(r,\theta)') T;@;R%
% K/A*<<r
~
% Example 2: $}lbT15a
% N5* u]j
% % Display the first 10 Zernike functions hZh9uI7.
% x = -1:0.01:1; mu?Eco`~
% [X,Y] = meshgrid(x,x); x8Retuv
% [theta,r] = cart2pol(X,Y); b|cyjDMAA
% idx = r<=1; $wmvKQc{lx
% z = nan(size(X)); .gG1kW A-
% n = [0 1 1 2 2 2 3 3 3 3]; 350_CN,
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; n3}!p'-CC
% Nplot = [4 10 12 16 18 20 22 24 26 28]; @Gx.q&H
% y = zernfun(n,m,r(idx),theta(idx)); wSb1"a
% figure('Units','normalized') U Z.=aQ}M
% for k = 1:10 8aO~/i:(.
% z(idx) = y(:,k); $Z|ffc1
% subplot(4,7,Nplot(k)) b'J'F;zh>
% pcolor(x,x,z), shading interp D@.tkzU@E
% set(gca,'XTick',[],'YTick',[]) 1"/He ` 4
% axis square A/s>PhxV
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) { T4
% end e_s&L,ze
% #[zI5)Meh
% See also ZERNPOL, ZERNFUN2. \]P!.}nX#
&8%e\W\K:/
% Paul Fricker 11/13/2006 V6t,BJjS
Vl_:c75"
GytXFL3`:
% Check and prepare the inputs: -:30:oq
% ----------------------------- 43={Xy
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |~'IM3Jw(Y
error('zernfun:NMvectors','N and M must be vectors.') GDu~d<R H
end P`#Z9 HM4
L,mQ
if length(n)~=length(m) [F*.\
error('zernfun:NMlength','N and M must be the same length.') '|S%aMLZ)
end [[>wB[w
*H?!;u=8
n = n(:); $-#Yl&?z9
m = m(:); U>V&-kxtV
if any(mod(n-m,2)) \2ZPj)&-E
error('zernfun:NMmultiplesof2', ... ?*?RP)V
'All N and M must differ by multiples of 2 (including 0).') A,\6nO67
end Fx5d:!]:$?
y]J89
if any(m>n) {]E+~%Va
error('zernfun:MlessthanN', ... K$ M^gh0
'Each M must be less than or equal to its corresponding N.') N@O8\oQG
end L:_bg8eD#
6)vSG7Ise
if any( r>1 | r<0 ) L3G \
error('zernfun:Rlessthan1','All R must be between 0 and 1.') PQK(0iCo4
end ]4R[<<hd
\e!vj.PU
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) z "+Mrew
error('zernfun:RTHvector','R and THETA must be vectors.') GP&vLt51
end r *$Ner
Z^]|o<.<I
r = r(:); {e+-vl
theta = theta(:); 1Ab>4UhD
length_r = length(r); OiE;B
if length_r~=length(theta) -RS7h
error('zernfun:RTHlength', ... \0mb
3Q'
'The number of R- and THETA-values must be equal.') ;=<-5;rI
end 'v\L @"
"Kc>dJ@W
% Check normalization: RjWqGr;bO
% -------------------- :$_6SQ<?
if nargin==5 && ischar(nflag) :=8t"rO=W
isnorm = strcmpi(nflag,'norm'); J?Dq>%+^
if ~isnorm j'aHF#_
error('zernfun:normalization','Unrecognized normalization flag.') LwhyE:1
end )ZBY* lk9
else E\IlF 6
isnorm = false; H(Q.a=&4!p
end *;m5'}jsy
OM|Fwr$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F29va
% Compute the Zernike Polynomials 'yV?*a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -0_d/'d
K
=wBpLB
% Determine the required powers of r: sf]s",t~J
% ----------------------------------- c\ia6[3sX
m_abs = abs(m); hSK;V<$[Z
rpowers = []; rQEyD
for j = 1:length(n) RPIyO
rpowers = [rpowers m_abs(j):2:n(j)]; C=s1R;"H
end aB]m*~
rpowers = unique(rpowers); $b<6y/"
k51Eyy50(
% Pre-compute the values of r raised to the required powers, <`jLY)sw
% and compile them in a matrix: ,(.MmP`
% ----------------------------- <L{(Mj%Z
if rpowers(1)==0 _[Vf547vS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P ~#>H{
rpowern = cat(2,rpowern{:}); 8a_[B~
rpowern = [ones(length_r,1) rpowern]; 8[|UgI,>z
else i~3u>CT
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Gcb|W&
rpowern = cat(2,rpowern{:}); eL4NB$Fb
end {tThy#
-F=v6N {
% Compute the values of the polynomials: }?&k a$rI
% -------------------------------------- P
i Fm|
y = zeros(length_r,length(n)); +3a?`Z
for j = 1:length(n) C-8qj>
s = 0:(n(j)-m_abs(j))/2; h Xb%;GL
pows = n(j):-2:m_abs(j); n!')wIk
for k = length(s):-1:1 K9vIm4::d$
p = (1-2*mod(s(k),2))* ... Qj3a_p$)P
prod(2:(n(j)-s(k)))/ ... xl"HotsX-x
prod(2:s(k))/ ... D;I6Q1I
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cgb2K$B_"
prod(2:((n(j)+m_abs(j))/2-s(k))); 'S[++w?Qq
idx = (pows(k)==rpowers); @]qBF]6
y(:,j) = y(:,j) + p*rpowern(:,idx); +4\U)Z/\
end S}f?.7
DAwqo.m
if isnorm s;1]tD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |A%<Z(
end }gkM^*$:%
end Hg9CZMko
% END: Compute the Zernike Polynomials JT9N!CGZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?=VOD #)
pA;-vMpMj
% Compute the Zernike functions: v4RlLgdS%
% ------------------------------ hky;CD~$
idx_pos = m>0; ork=`};
idx_neg = m<0; T~fmk
f$
[xh*"wT#g
z = y; 4lqH8l.
if any(idx_pos) a=XW[TY1
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }|B=h
end SxK:]Aw
if any(idx_neg) ~2d:Q6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?:|-Dq,
end }n7th
m%"uPv\
% EOF zernfun