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 <ShA_+Nd
function z = zernfun(n,m,r,theta,nflag) me{u~9&
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. uoOUgNwGg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N xpO;V}M|
% and angular frequency M, evaluated at positions (R,THETA) on the +&S6se4
% unit circle. N is a vector of positive integers (including 0), and [>r0
(x&.
% M is a vector with the same number of elements as N. Each element `Fo/RZOW
% k of M must be a positive integer, with possible values M(k) = -N(k) 4bp})>}jB
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \lm]G7h
% and THETA is a vector of angles. R and THETA must have the same fqY'Uq$=
% length. The output Z is a matrix with one column for every (N,M) 4oH ,_sr
% pair, and one row for every (R,THETA) pair. :UP8nq
% /5/gnpC
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z'$1$~I
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =EMB~i
% with delta(m,0) the Kronecker delta, is chosen so that the integral }mK,Bi?bj
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aA52Li
% and theta=0 to theta=2*pi) is unity. For the non-normalized 6
iMJ0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5qW>#pTFVV
% A9 g%>
% The Zernike functions are an orthogonal basis on the unit circle. ]uypi#[
% They are used in disciplines such as astronomy, optics, and YS){N=g&'
% optometry to describe functions on a circular domain. .?Y"o3
% xlJWCA*>
% The following table lists the first 15 Zernike functions. &Q;sbI}
% ~=iH*AQR
% n m Zernike function Normalization CX{6
% -------------------------------------------------- Dqii60
% 0 0 1 1 D?"P\b[/
% 1 1 r * cos(theta) 2 .kg 3>*
% 1 -1 r * sin(theta) 2 <7F-WR/2n
% 2 -2 r^2 * cos(2*theta) sqrt(6) aK
-x{
% 2 0 (2*r^2 - 1) sqrt(3) B+U:=591
% 2 2 r^2 * sin(2*theta) sqrt(6) ^7gKs2M
% 3 -3 r^3 * cos(3*theta) sqrt(8) oC49c~`8
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]#^v754X^T
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) rG6G~|mS
% 3 3 r^3 * sin(3*theta) sqrt(8) _Iav2=0Wi
% 4 -4 r^4 * cos(4*theta) sqrt(10) gee~>l
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?..BA&zRk
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) th[v"qD9G
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t~j6w sx;
% 4 4 r^4 * sin(4*theta) sqrt(10) $1|E(d1
% -------------------------------------------------- bV&9>fC
% ` qs}L
% Example 1: r4X}U|s!0
% \8QOZjy
% % Display the Zernike function Z(n=5,m=1) .cQO?UKK
% x = -1:0.01:1; <JWU@A-.y
% [X,Y] = meshgrid(x,x); jBYvOy*$Q
% [theta,r] = cart2pol(X,Y); v;o1c44;
% idx = r<=1; pN5kcvQ
% z = nan(size(X)); 2vjkThh`I
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~W8Xg)
% figure >lUPOc
% pcolor(x,x,z), shading interp "nu]3zcd
% axis square, colorbar ;un@E:
% title('Zernike function Z_5^1(r,\theta)') S
\]O8#OX
% "4\
% Example 2: EwN{| 34C
% h>\C2Q
% % Display the first 10 Zernike functions s<F*kLib
% x = -1:0.01:1; d'ZNp2L
% [X,Y] = meshgrid(x,x); j@z IJ
% [theta,r] = cart2pol(X,Y); Mww ^
% idx = r<=1; /Rq\Mgb
% z = nan(size(X)); >pfeP"[(3
% n = [0 1 1 2 2 2 3 3 3 3]; K9k!P8Rd
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~ h3G}EH
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {V
QGfN
% y = zernfun(n,m,r(idx),theta(idx)); ]A=\P,D
% figure('Units','normalized') OA3J(4!"W
% for k = 1:10 -[-oz0`Sl{
% z(idx) = y(:,k); (V6bX]<
% subplot(4,7,Nplot(k)) BjvQ6M{Y"+
% pcolor(x,x,z), shading interp SKH}!Id}n
% set(gca,'XTick',[],'YTick',[]) (^}t
% axis square JK =A=
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]64}Xob87_
% end c}qpmW F
% V)<>W_g
% See also ZERNPOL, ZERNFUN2. ,]2?S5R
c{/R?<
% Paul Fricker 11/13/2006 n]IF`kYQV
dRJ
](Gw
XMI*obS'z
% Check and prepare the inputs: /@ @F
nQ++
% ----------------------------- n;Oe- +oSC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dw <i)P^
error('zernfun:NMvectors','N and M must be vectors.') s0?'mC+p
end DPzW,aIgv
rV%68x9
if length(n)~=length(m) C{J5:ak
error('zernfun:NMlength','N and M must be the same length.') hUlRtt
end
AfTm#-R
et
1HbX
n = n(:); o7!A(Eu
m = m(:); IEy$2f>Ns
if any(mod(n-m,2)) 3$!QP
N
error('zernfun:NMmultiplesof2', ... dA hcA.
'All N and M must differ by multiples of 2 (including 0).') }) -V,\
end y]jx-wc3O
6LDZ|K@
if any(m>n) iP(MDVg
error('zernfun:MlessthanN', ... ~DK.Y
'Each M must be less than or equal to its corresponding N.') 5qnei\~
end plWNuEW
#,#_"
if any( r>1 | r<0 ) 8?nn4]P
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z2]0brV
end FFw(`[A_
.:j{d}p}
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XS&Pc
error('zernfun:RTHvector','R and THETA must be vectors.') 8UjIC4'
end w PR Ns9^
\XB,)XDB
r = r(:); X
<xM '
theta = theta(:); 8`*5[ L~~/
length_r = length(r); 1-p#}VX
if length_r~=length(theta) #a}w&O";
error('zernfun:RTHlength', ... -KGJr
'The number of R- and THETA-values must be equal.') M$EF 8
end m-O*t$6
t`JT
% Check normalization: PL=v,NB
% -------------------- ^`yhN
if nargin==5 && ischar(nflag) bDvGFSAH
isnorm = strcmpi(nflag,'norm'); U^7hw(}me
if ~isnorm ~},H+A!?
error('zernfun:normalization','Unrecognized normalization flag.') AJ/Hw>>$?m
end %_E5B6xi{
else pA .orx
isnorm = false; ^N<aHFF
end GhfhR^P
.$-;`&0cZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YeOn
% Compute the Zernike Polynomials !6|_`l>G,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2*D2jw
\5}PF+)|
% Determine the required powers of r: 1^$hbRq
% ----------------------------------- Q I";[
m_abs = abs(m); hXI[FICQU{
rpowers = []; ZiR}S
for j = 1:length(n) _(f@b1O~
rpowers = [rpowers m_abs(j):2:n(j)]; $CB&>?~
end h's[)
t
rpowers = unique(rpowers); ]xvhUv!G
l#cVQ_^"
% Pre-compute the values of r raised to the required powers, P7}w^#x
% and compile them in a matrix: :j+E]|d(~6
% ----------------------------- \)28,`
if rpowers(1)==0 *=@8t^fa86
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ek)rsxf1A
rpowern = cat(2,rpowern{:}); GThGV"
rpowern = [ones(length_r,1) rpowern]; Q3ZGN1aX<
else kVtP~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E~q3o*
rpowern = cat(2,rpowern{:}); \_.'/<aQ
end yzfiH4
;VCV%=W<
% Compute the values of the polynomials: 1<@lM8&.kO
% -------------------------------------- Lb$Uba-_
y = zeros(length_r,length(n)); s8(Z&pQ
for j = 1:length(n) hRuiuGC
s = 0:(n(j)-m_abs(j))/2; ZOqA8#\
pows = n(j):-2:m_abs(j); ^e "4@O"
for k = length(s):-1:1 jR1^e$
p = (1-2*mod(s(k),2))* ... AIl`>ac
prod(2:(n(j)-s(k)))/ ... rMG[,:V
prod(2:s(k))/ ... W9gQho%9b
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... u^C\aujg
prod(2:((n(j)+m_abs(j))/2-s(k))); >}.~Y#Ge
idx = (pows(k)==rpowers); XKp(31])
y(:,j) = y(:,j) + p*rpowern(:,idx); EO'+r[Y
end 2O(k@M5E?
TS=%iMa
if isnorm gz'{l[
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \l(}8;5}
end si%V63 ^lN
end T:Q+ Z }v+
% END: Compute the Zernike Polynomials q:vN3#=^qf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fc:87ZR{K
6/QWzw.0c
% Compute the Zernike functions: w2 (}pz:
% ------------------------------ .nr%c*JUp
idx_pos = m>0; ?>=vKU5
idx_neg = m<0; 0* ^f
EoV
svo%NQ
z = y; ,EH-Sf2Cb
if any(idx_pos) zGO_S\
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #/(L.5d[
end pkIQ,W{Ke
if any(idx_neg) tm34Z''.>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +7"UF)
~k
end B$=1@
/;TD n>lq
% EOF zernfun