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 OjeM#s#N!
function z = zernfun(n,m,r,theta,nflag) [>?B`1;@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^O[qCX
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N wTIOCj
% and angular frequency M, evaluated at positions (R,THETA) on the ;Cyt2]F
% unit circle. N is a vector of positive integers (including 0), and S]{K^Q),
% M is a vector with the same number of elements as N. Each element eVbHPu4
% k of M must be a positive integer, with possible values M(k) = -N(k) :fpYraBM
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, AytHnp\H
% and THETA is a vector of angles. R and THETA must have the same I#S6k%-'
% length. The output Z is a matrix with one column for every (N,M) g4j?E{M?
% pair, and one row for every (R,THETA) pair. U4zyhj
% aCFO]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike iu&'v
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 't^OIil
% with delta(m,0) the Kronecker delta, is chosen so that the integral P7"g/j" "
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, > -Jd@7-
% and theta=0 to theta=2*pi) is unity. For the non-normalized
; >.>vLF
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7PP76$
% 01#a
% The Zernike functions are an orthogonal basis on the unit circle. ep,kImT
% They are used in disciplines such as astronomy, optics, and jcOxtDTSW
% optometry to describe functions on a circular domain. LYavth`@h
% (?YTQ8QR
% The following table lists the first 15 Zernike functions. sRb)*p'
% 0P\)L`cG
% n m Zernike function Normalization )MW.Y
% -------------------------------------------------- :)?w2'O
% 0 0 1 1 E@P8-x'i
% 1 1 r * cos(theta) 2 hq$:62NYg
% 1 -1 r * sin(theta) 2 [ZOo%"M_Y
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ry[VEn>C1
% 2 0 (2*r^2 - 1) sqrt(3) JyYg)f
% 2 2 r^2 * sin(2*theta) sqrt(6) RP z0WP
% 3 -3 r^3 * cos(3*theta) sqrt(8) srJ,Jr(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *V3 }L
Z
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) +!dIEt).U
% 3 3 r^3 * sin(3*theta) sqrt(8) mTYEK4}
% 4 -4 r^4 * cos(4*theta) sqrt(10) "F}anPY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0b~5i-zM/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 6
}qNH29
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?fc({zb
% 4 4 r^4 * sin(4*theta) sqrt(10) L5of(gQ5]
% -------------------------------------------------- ft4J.oT
% B.;/N220P
% Example 1: D*DCMMp=0
% XNf%vC>
% % Display the Zernike function Z(n=5,m=1) :_i1)4[!
% x = -1:0.01:1; %{5mkO&,2
% [X,Y] = meshgrid(x,x); @q],pD
% [theta,r] = cart2pol(X,Y); S;a{wYF6v
% idx = r<=1; 9eH(FB
% z = nan(size(X)); $^y6>@~
% z(idx) = zernfun(5,1,r(idx),theta(idx)); e
,k,L
% figure ,57g_z]V
% pcolor(x,x,z), shading interp IdUMoLL?
% axis square, colorbar y 7|x<Z
% title('Zernike function Z_5^1(r,\theta)') DL_2%&k/
% |u<qbl
% Example 2: j$n[;\]n
% FG38) /
% % Display the first 10 Zernike functions
TfDx>
F$
% x = -1:0.01:1; pZuYmMP
% [X,Y] = meshgrid(x,x); aRC>pK.
% [theta,r] = cart2pol(X,Y); oXK`=.\
% idx = r<=1; Se%FqI
% z = nan(size(X)); Gyk>5Q}}
% n = [0 1 1 2 2 2 3 3 3 3]; nUz2~z
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ; mu9;ixZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *Ny^XQ_ X
% y = zernfun(n,m,r(idx),theta(idx)); Ef;_im
% figure('Units','normalized') #|
`W ]
% for k = 1:10 3YR6@*!f/
% z(idx) = y(:,k); =oV8!d%]
% subplot(4,7,Nplot(k)) c1'OIK C
% pcolor(x,x,z), shading interp sFC&DTb?
% set(gca,'XTick',[],'YTick',[]) iKu[j)F
% axis square 7,d^?.~S
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) KCGs*kp>
% end sf2_x>U1
% zT6ng#
% See also ZERNPOL, ZERNFUN2. BBm.;=8@ ^
-P]J:7*0?\
% Paul Fricker 11/13/2006 $M@SZknm
{l-,Jbfi`
- (VV
% Check and prepare the inputs: muwXzN(KX
% ----------------------------- 1c(1 YGuH
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4r\Sbh
error('zernfun:NMvectors','N and M must be vectors.') Pwt4e-
end f9cS^v_:
>r2m1}6g"
if length(n)~=length(m) '""qMRCm
error('zernfun:NMlength','N and M must be the same length.') kZs
end &n|#jo(gS
..X efNbl
n = n(:); %qcBM~efT
m = m(:); =#[_8)q
if any(mod(n-m,2)) GrGgR7eC#P
error('zernfun:NMmultiplesof2', ... +[V[{n
'All N and M must differ by multiples of 2 (including 0).') ^)m]j`}IGb
end i DO`N!
2T<QG>;)j
if any(m>n)
Ash"D~
error('zernfun:MlessthanN', ... 8)&H=#E
'Each M must be less than or equal to its corresponding N.') Z~F% K~(
end
S
U~vS
%f\ M61Z
if any( r>1 | r<0 ) .^N+'g
error('zernfun:Rlessthan1','All R must be between 0 and 1.') s[)2z3
end =i~/.Nu&
W@GcE;#-
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) v)N8vFdd
error('zernfun:RTHvector','R and THETA must be vectors.') [-bL>8
end 6*Qn9Q%p-
X&0m$x
r = r(:); 6cp x1y]~6
theta = theta(:); `9B xDp]I
length_r = length(r); _tS<\zy@y
if length_r~=length(theta) eC%.xu^
error('zernfun:RTHlength', ... $74ZC
M
'The number of R- and THETA-values must be equal.') @Ytsb!!
end j9XY%4.
g-U'{I5F
% Check normalization: Pk T&zSQA
% -------------------- L;I.6<K.
if nargin==5 && ischar(nflag) )p4o4aM
isnorm = strcmpi(nflag,'norm'); Hq8<g$
if ~isnorm R!lNm,i
error('zernfun:normalization','Unrecognized normalization flag.') &_HSrU
end =\e}fyuK
else Y=sRVypJ
isnorm = false; HUr;ysw
end b[$%Wg
Vj_(55WQ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8s-RNA>7^
% Compute the Zernike Polynomials k$y(H;XA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wz nz
x~!B.4gT2
% Determine the required powers of r: S&}7jRH1
% ----------------------------------- cv eTrY}g
m_abs = abs(m); [Tby+pC
rpowers = []; m>k
j @^SQ
for j = 1:length(n) {J%Na&D
rpowers = [rpowers m_abs(j):2:n(j)]; E
`Ualai
end I7r{&X) D
rpowers = unique(rpowers); "B*a|
'n!
n9]^v-]K
% Pre-compute the values of r raised to the required powers,
B]ul~FX
% and compile them in a matrix: 7f8%WD)
% ----------------------------- [@U8&W
if rpowers(1)==0 f)H6 nl7r
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~0w7E0DE[
rpowern = cat(2,rpowern{:}); *#;8mM
rpowern = [ones(length_r,1) rpowern]; N(vzxx^
else C6,GgDH`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'C+z
rpowern = cat(2,rpowern{:}); #HWz.Wb
end W:O<9ZbQ_
QG?7L_I
% Compute the values of the polynomials: DalQ.
% -------------------------------------- t1b$,jHmKl
y = zeros(length_r,length(n)); *_`T*$
for j = 1:length(n) `J[(Dx'y=t
s = 0:(n(j)-m_abs(j))/2; j WLZ!a3+
pows = n(j):-2:m_abs(j); @^a6^*X>
for k = length(s):-1:1 (9*s:)zD-
p = (1-2*mod(s(k),2))* ... 0&=2+=[c
prod(2:(n(j)-s(k)))/ ... z{pNQ[t1Z
prod(2:s(k))/ ... ^c83_93)R
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... sSd
prod(2:((n(j)+m_abs(j))/2-s(k))); +P/"bwv0
idx = (pows(k)==rpowers); ;38W41d{
y(:,j) = y(:,j) + p*rpowern(:,idx); %1gJOV
end a3?Dtoy'
Q-F'-@`(C
if isnorm 9Re605xQ6
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kQC>8"
end fU@}]&
end RKdf1C
% END: Compute the Zernike Polynomials 7loCb4Hv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ky|Hi3?
GC66n1- X
% Compute the Zernike functions: }r]WB)_w
% ------------------------------ %\_I%
yF
idx_pos = m>0; Z{+h~?63
idx_neg = m<0; t!c8c^HR
JmrQDO_(
z = y; 8xj4N%PA
if any(idx_pos) }U7>_b2
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); B
h@R9O<
end Ox?LVRvxI
if any(idx_neg) #jd?ocoY
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); YH)Unql
end j8zh^q
vF;6Y(h>
% EOF zernfun