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 umj7-fh
function z = zernfun(n,m,r,theta,nflag) ){/y-ixH
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. dW91nTQ:
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6w5 4+n
% and angular frequency M, evaluated at positions (R,THETA) on the N$.''D?7D
% unit circle. N is a vector of positive integers (including 0), and edm&,ph]
% M is a vector with the same number of elements as N. Each element X|b~,X%N
% k of M must be a positive integer, with possible values M(k) = -N(k) 2'++G[z
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _A(J^;?
% and THETA is a vector of angles. R and THETA must have the same FQ[::*-
% length. The output Z is a matrix with one column for every (N,M) 1m&(3%#{
% pair, and one row for every (R,THETA) pair. .DT1Jvl
% UOq$88sr
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike g{&ux k);
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,Ti#g8j
% with delta(m,0) the Kronecker delta, is chosen so that the integral oo2VT
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ";Lpf]<
% and theta=0 to theta=2*pi) is unity. For the non-normalized Qv8Z64#
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. K@hv[4
% 3ZC[H'|
% The Zernike functions are an orthogonal basis on the unit circle. J,k{Bm
% They are used in disciplines such as astronomy, optics, and K }r%OOn0
% optometry to describe functions on a circular domain. q4VOK
'N
% b
afYjF< 3
% The following table lists the first 15 Zernike functions. S\Q/ "Y
% o zv><e#
% n m Zernike function Normalization d6_ CsqV
% -------------------------------------------------- "g0Ln5&
% 0 0 1 1 iNha<iS+
% 1 1 r * cos(theta) 2 |d8/ZD
% 1 -1 r * sin(theta) 2 {BgGG@e
% 2 -2 r^2 * cos(2*theta) sqrt(6) R#gip
% 2 0 (2*r^2 - 1) sqrt(3) #[2]B8NZ
% 2 2 r^2 * sin(2*theta) sqrt(6) &B[$l`1
% 3 -3 r^3 * cos(3*theta) sqrt(8) Z$T1nm%lo:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) #b:8-Lt:M
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fAJQ8nb{@]
% 3 3 r^3 * sin(3*theta) sqrt(8) a(bgPkPP
% 4 -4 r^4 * cos(4*theta) sqrt(10) NoV2<m$
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @ %kCe>r
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) iN_G|w[d
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #;H+Kb5O
% 4 4 r^4 * sin(4*theta) sqrt(10) T-eeYw?Yf
% -------------------------------------------------- 7kHEY5s
"
% i9_ZK/*
% Example 1: nx=Zl:Q}
% w$pBACX
% % Display the Zernike function Z(n=5,m=1) ?DA,]aa-
% x = -1:0.01:1; :v=Yo
% [X,Y] = meshgrid(x,x); )
=sm{R%T
% [theta,r] = cart2pol(X,Y);
|G{TA
% idx = r<=1; *l^h;RSx
% z = nan(size(X)); ?> }bg
% z(idx) = zernfun(5,1,r(idx),theta(idx)); C;M.dd
% figure (@~d9PvB>
% pcolor(x,x,z), shading interp dtr8u
% axis square, colorbar YcT!`B
% title('Zernike function Z_5^1(r,\theta)') RD<l<+C^~
% lQY?!oj&q
% Example 2: //Ck1cI#h
% h`,dg%J*B
% % Display the first 10 Zernike functions a6fMx~
% x = -1:0.01:1; +U%
=
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% [X,Y] = meshgrid(x,x); $Ic:
c
% [theta,r] = cart2pol(X,Y); Xh;Pbm|K
% idx = r<=1; 94LFElE3
% z = nan(size(X)); ._Wm%'uX
% n = [0 1 1 2 2 2 3 3 3 3]; \XD&0inv
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; )k{zRq:d
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 1
@tVfn}
% y = zernfun(n,m,r(idx),theta(idx)); \|R P-8
% figure('Units','normalized') G ,An8GR%&
% for k = 1:10 !0{":4\
% z(idx) = y(:,k); w-pdpbHV
% subplot(4,7,Nplot(k)) }hv>LL
% pcolor(x,x,z), shading interp .|;`qUo
% set(gca,'XTick',[],'YTick',[]) .-Ggvw
% axis square *^ g7kCe(
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;"Q{dOvp
% end VD#`1g<
% +h.$<=
% See also ZERNPOL, ZERNFUN2. !O~EIz
p
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% Paul Fricker 11/13/2006 MS)(\&N
a39Kl_\
AtGk
_tpVZ
% Check and prepare the inputs: @.6l^"L
% ----------------------------- B0T[[%~3M
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [/.o>R#J(
error('zernfun:NMvectors','N and M must be vectors.') -Xb]=Yf-
end hlWTsi4N
wz3BtCx
if length(n)~=length(m) 3@f@4t@5V
error('zernfun:NMlength','N and M must be the same length.') Zu951+&`
end LS}dt?78`V
6lpfk&
n = n(:); 4{7O}f
m = m(:); GcmN40
if any(mod(n-m,2)) v,#*%Gn`%
error('zernfun:NMmultiplesof2', ... yS%IE>?
'All N and M must differ by multiples of 2 (including 0).') -SnP+X!
end n$i}r\
so
J39,x=8LL
if any(m>n) 8wKF.+_A
error('zernfun:MlessthanN', ... ),1MR=
'Each M must be less than or equal to its corresponding N.') c4E=qgP
end uU=O 0?'zq
zZE
2%fqM
if any( r>1 | r<0 ) l$=Y(Xk
error('zernfun:Rlessthan1','All R must be between 0 and 1.') R]L|&{
end 7vax[,aI
3C{3"bP
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) wyvrNru<l4
error('zernfun:RTHvector','R and THETA must be vectors.') H48`z'o
end ZbD_AP
ve;#o<
r = r(:); zBg>I=hiG
theta = theta(:); \x\_I1|
length_r = length(r); 2A'!kd$2
if length_r~=length(theta) k \rzvo=U
error('zernfun:RTHlength', ... "$#X[.
'The number of R- and THETA-values must be equal.') !l-^JPb
end
?UuJk
2YI#J.6]H
% Check normalization: 5RD\XgyN]
% -------------------- #
Un>g4>Rh
if nargin==5 && ischar(nflag) tp"dho
isnorm = strcmpi(nflag,'norm'); Ad !=
*n
if ~isnorm *Y(v!x \L
error('zernfun:normalization','Unrecognized normalization flag.') IMjz#|c
end #/!fLU@
else hqOy*!8'@
isnorm = false; rjqQWfShY
end (:v|(Gn/
{YnR]|0&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&0! f_
% Compute the Zernike Polynomials /cM<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G
Xx7/ X
xrb %-vT
% Determine the required powers of r: r:Uqtqxh
% ----------------------------------- l,5<g-r
V
m_abs = abs(m); m:U.ao6
rpowers = []; {nTQc2T?;
for j = 1:length(n) xdw"JS}
rpowers = [rpowers m_abs(j):2:n(j)]; V8AF;1c?-'
end Sz4G,c
rpowers = unique(rpowers); M\\t)=q
pt[H5
% Pre-compute the values of r raised to the required powers, i
Lr*W#E
% and compile them in a matrix: #m
yiZL%
% ----------------------------- z/09~Hc
if rpowers(1)==0 k+Ew+j1_
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); n/*BK;
rpowern = cat(2,rpowern{:}); mHcxK@qw
rpowern = [ones(length_r,1) rpowern]; 1 ?X(q
else .<uxZ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ::bK{yZm
rpowern = cat(2,rpowern{:}); Hjl{M>z
end uFxhr2
<z
?S&pq?
% Compute the values of the polynomials: LS1r}cl
% -------------------------------------- :6R0=oz
y = zeros(length_r,length(n)); 2ZHeOKJ-
for j = 1:length(n) ia=eFWt.
s = 0:(n(j)-m_abs(j))/2; OT-!n
pows = n(j):-2:m_abs(j); Np$peT[
for k = length(s):-1:1 l"9.zPvT<
p = (1-2*mod(s(k),2))* ... Fh t$7V
prod(2:(n(j)-s(k)))/ ... =fA*b
prod(2:s(k))/ ... -)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *]uo/g
prod(2:((n(j)+m_abs(j))/2-s(k))); K5X,J/n
idx = (pows(k)==rpowers); NR3]MGBKv
y(:,j) = y(:,j) + p*rpowern(:,idx); S<),
,(
end $gKMVgD"
#H]b Xr
if isnorm dV+%x"[:
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1O" Mo
end #XSs.i{
end s-^B)0T!
% END: Compute the Zernike Polynomials oq00)I1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y k=o
hCXSC*;
% Compute the Zernike functions: }~gBnq_DDU
% ------------------------------ L0ZgxG3:g
idx_pos = m>0; ~~J xw ]
idx_neg = m<0; rKZ1
c,y
GL4-v[]6I
z = y; P_:A%T
if any(idx_pos) {O\>"2}m'f
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "&jWC
end ziFg+i%s
if any(idx_neg) N^,@s"g
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); P}=u8(u
end a%3V<
"f
B+e$S%HV
% EOF zernfun