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 vccWe7rh
function z = zernfun(n,m,r,theta,nflag) BEZ~<E&0H
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !\]^c
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,RP-)j"Wff
% and angular frequency M, evaluated at positions (R,THETA) on the R^Rc!G}
% unit circle. N is a vector of positive integers (including 0), and c=\tf~}^Ms
% M is a vector with the same number of elements as N. Each element ^Fk;t
% k of M must be a positive integer, with possible values M(k) = -N(k) [ X*p
[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 6*8Wtq
% and THETA is a vector of angles. R and THETA must have the same LvG.ocCG
% length. The output Z is a matrix with one column for every (N,M) a+h$u
% pair, and one row for every (R,THETA) pair. wNONh`b
% }v1wpv/b(
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;!yK~OBxt
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), bT,:eA
% with delta(m,0) the Kronecker delta, is chosen so that the integral FU|brSt
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, w+o5iPLX
% and theta=0 to theta=2*pi) is unity. For the non-normalized =;Id["+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. PSrx!
% _ %s#Cb
% The Zernike functions are an orthogonal basis on the unit circle. W?7l-k=S
% They are used in disciplines such as astronomy, optics, and ~C-Sr@ a?/
% optometry to describe functions on a circular domain. uf(ayDE
% P\7DA4]
% The following table lists the first 15 Zernike functions. S :HOlJze
% Ht`fC|E
% n m Zernike function Normalization 5zuwqOD*
% -------------------------------------------------- 2Gyq40
% 0 0 1 1 ~NGM6+9
% 1 1 r * cos(theta) 2 l,ny=Q$[1'
% 1 -1 r * sin(theta) 2
l\U
Q2i
% 2 -2 r^2 * cos(2*theta) sqrt(6) 1-RY5R}VR
% 2 0 (2*r^2 - 1) sqrt(3) j*=!M# D
% 2 2 r^2 * sin(2*theta) sqrt(6) dQX-s=XJ
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^[ae
)}
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) verI~M$v{
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) +/OSg.
% 3 3 r^3 * sin(3*theta) sqrt(8)
w7)pBsI
% 4 -4 r^4 * cos(4*theta) sqrt(10) I2}W /}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N,t9X7G&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) KbJ6U75|f
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rcnH ^P
% 4 4 r^4 * sin(4*theta) sqrt(10) bC&A@.g{
% -------------------------------------------------- 1nVQYqT_
% ]l7W5$26 @
% Example 1: "tEp8m
% lH fZw})d
% % Display the Zernike function Z(n=5,m=1) +Z#=z,.^
% x = -1:0.01:1; FlO?E3d
% [X,Y] = meshgrid(x,x); SX3'|'-
% [theta,r] = cart2pol(X,Y); EPo)7<|>
% idx = r<=1; 8)B{x[?|
% z = nan(size(X)); X)g
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); A#>wbHjWF
% figure ]+lT*6P*
% pcolor(x,x,z), shading interp D@=]mh6vl
% axis square, colorbar VPCI5mS_
% title('Zernike function Z_5^1(r,\theta)') =^"Sx??V
% Q0*E&;|
% Example 2: v gW(l2,@
% hvt]VC]]
% % Display the first 10 Zernike functions \Y#
% x = -1:0.01:1; MmJMx
% [X,Y] = meshgrid(x,x); .0Ud?v>=
% [theta,r] = cart2pol(X,Y); _/[qBe
% idx = r<=1; s>9I#_4]
% z = nan(size(X)); :?f<tNU$
% n = [0 1 1 2 2 2 3 3 3 3]; )L<.;`g4x
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !D22HSv(w
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 6v@Prw@.b
% y = zernfun(n,m,r(idx),theta(idx)); 0jp].''RK\
% figure('Units','normalized') <3Ftq=
% for k = 1:10 v]JET9hY
% z(idx) = y(:,k); >^8O :.
% subplot(4,7,Nplot(k)) Rsx6vF8]5
% pcolor(x,x,z), shading interp aru2H6
% set(gca,'XTick',[],'YTick',[]) _ep&`K
% axis square o!xCM:+J
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) jMT[+f
% end ? [Yn<|
% 6O4*OR<&
% See also ZERNPOL, ZERNFUN2. }3
/io0"D
p{?duq=
% Paul Fricker 11/13/2006 V``|<`!gd
GTs,?t16/
{\Pk;M{Y&
% Check and prepare the inputs: 5%'ybh)@
% ----------------------------- -6MPls+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) AA-$;s
error('zernfun:NMvectors','N and M must be vectors.') 4'faE="1)S
end %
:G78.
h(WlJCln
if length(n)~=length(m) e`Yj}i*bx]
error('zernfun:NMlength','N and M must be the same length.') 8YSvBy
end qMaO1cE\
,|f=2t+5X
n = n(:); 8;M,l2pmR{
m = m(:); e_-g|ukC
if any(mod(n-m,2)) #kQ! GMZH
error('zernfun:NMmultiplesof2', ... n3e,vP? R
'All N and M must differ by multiples of 2 (including 0).') e"@r[pq-{u
end q~>!_q]FE
zDg*ds\
if any(m>n) R/u0,
error('zernfun:MlessthanN', ... 4n#u?)
'Each M must be less than or equal to its corresponding N.') mjOxmwo
end {UH45#Ua
?`TQ!m6y
if any( r>1 | r<0 ) ]xf89[;0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') :F d1k
Jm
end QXI~Toddj
Eq@sU?j
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I/'>MDB!
error('zernfun:RTHvector','R and THETA must be vectors.') b$w66q8
end 28JVW3&)
*wAX&+);
r = r(:); +sJ{9# 6
theta = theta(:); tE>FL
length_r = length(r); -raK
if length_r~=length(theta) oD%n}
error('zernfun:RTHlength', ... NO/$}vw
'The number of R- and THETA-values must be equal.') C,,T7(: k
end ?Gf'G{^}
:qS~"@ ?<
% Check normalization: bLTX_
R
% -------------------- +:m)BLA4l
if nargin==5 && ischar(nflag) /XG7M=A$o
isnorm = strcmpi(nflag,'norm'); j gV^{8qG
if ~isnorm TaF*ZT2
error('zernfun:normalization','Unrecognized normalization flag.') (9bU\4F\
end 5hqXMs
else DKo6lP`
isnorm = false; W)`>'X`
end :~ s"]*y
j % MY6"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VK9E{~0=
% Compute the Zernike Polynomials uP7|#>1%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e?\Od}Hbw
]Y
&
2&
% Determine the required powers of r: Y&VypZ"G>
% ----------------------------------- AU*]D@H
m_abs = abs(m); dyqk[$(
rpowers = []; HH*,Oe
for j = 1:length(n) :wzbD,/M
rpowers = [rpowers m_abs(j):2:n(j)]; YTgT2w
end =PU@'OG
rpowers = unique(rpowers); ( 3,7
$sL+k 'dY
% Pre-compute the values of r raised to the required powers, `U?S 9m
% and compile them in a matrix: aorL ,l
% ----------------------------- c5CxR#O
if rpowers(1)==0 lYS4Q`z$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bq7()ocA
rpowern = cat(2,rpowern{:}); *~`oA~-Q
rpowern = [ones(length_r,1) rpowern]; AED
9vDE
else w6 Y+Y;,'f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )W @
rpowern = cat(2,rpowern{:}); VQ~eg wJL
end EZ Q!~
?`*`A9@
% Compute the values of the polynomials: 4pDZ +}p
% -------------------------------------- U:/_T>f%
y = zeros(length_r,length(n)); ~9fTs4U
for j = 1:length(n) 4yu=e;C wy
s = 0:(n(j)-m_abs(j))/2; |bRi bB
pows = n(j):-2:m_abs(j); { F0"U=
for k = length(s):-1:1 hO3C _}
p = (1-2*mod(s(k),2))* ... xoSBMf
prod(2:(n(j)-s(k)))/ ... ;?o"{mbb
prod(2:s(k))/ ... F7p`zf@O]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8W.-Y|[5?
prod(2:((n(j)+m_abs(j))/2-s(k))); fQU_A
idx = (pows(k)==rpowers); RvW>kATb_F
y(:,j) = y(:,j) + p*rpowern(:,idx); ^-}3+YA
end +c'I7bBr
Tq6@
1j6p
if isnorm F,5}3$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 51%<N\>/4
end k/xNqN(
end [s!c c:JR
% END: Compute the Zernike Polynomials $L"-JNS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RML'C:1
ku5g`ho
% Compute the Zernike functions: U&tR1v'
% ------------------------------ TwE&5F*
idx_pos = m>0; "jl`FAu)q
idx_neg = m<0; H~qY7t
H%c{ }F
z = y; 0xutG/-&N
if any(idx_pos) 5a l44[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xeHqC9Ou
end 7w"YCRKh
if any(idx_neg) Kib?JRYt
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); q->46{s|
end Z$@ XMq!
@l2AL9z$m>
% EOF zernfun