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 Dck/Ea
function z = zernfun(n,m,r,theta,nflag) ?I?G+(bq
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. h,{Q%sqO
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `Na()r$T
% and angular frequency M, evaluated at positions (R,THETA) on the 5e7Y M@ng
% unit circle. N is a vector of positive integers (including 0), and _=$~l^Y[
% M is a vector with the same number of elements as N. Each element ^$ Y9.IH"
% k of M must be a positive integer, with possible values M(k) = -N(k) == wX.y\.n
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :$D*ab^^P
% and THETA is a vector of angles. R and THETA must have the same {N~mDUoJ|
% length. The output Z is a matrix with one column for every (N,M) &>qUT]w
% pair, and one row for every (R,THETA) pair. |:S6Gp[\O
% Aits<0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike CA#g(SiZ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {($bzT7c
% with delta(m,0) the Kronecker delta, is chosen so that the integral d.+*o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d[t0K]
% and theta=0 to theta=2*pi) is unity. For the non-normalized -`'|z+V
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. y11^q*}
% c~U0&V_`j
% The Zernike functions are an orthogonal basis on the unit circle. B?%u<F
% They are used in disciplines such as astronomy, optics, and ,g?ny<#o
% optometry to describe functions on a circular domain. nWsRauY
% b%lB&}uw}
% The following table lists the first 15 Zernike functions. 5.^pD9 [mT
% wKpGJ&
{
% n m Zernike function Normalization z<eu=OD4t
% -------------------------------------------------- P'VHga
% 0 0 1 1 l-w4E"n3
% 1 1 r * cos(theta) 2 <qR$ `mLN
% 1 -1 r * sin(theta) 2 $R}C(k
;?
% 2 -2 r^2 * cos(2*theta) sqrt(6) DG;u_6;JR
% 2 0 (2*r^2 - 1) sqrt(3) St?mq* ,
% 2 2 r^2 * sin(2*theta) sqrt(6) `:lcN0n
% 3 -3 r^3 * cos(3*theta) sqrt(8) Hs)]
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) c&T5C,]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wA{)9.
% 3 3 r^3 * sin(3*theta) sqrt(8) p+P@I7V
% 4 -4 r^4 * cos(4*theta) sqrt(10) &3/`cl[+
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .g/!u(iy
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) #xmiUN,|
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SsX$l<t*
% 4 4 r^4 * sin(4*theta) sqrt(10) wp:$Tq a$
% -------------------------------------------------- u0R[TA3
% %enJ[a%Qg
% Example 1: m@nGXl'!
% Yy
4Was#
% % Display the Zernike function Z(n=5,m=1) zpT{!V
% x = -1:0.01:1; 1%M^MT%&
% [X,Y] = meshgrid(x,x); >]}VD "\
% [theta,r] = cart2pol(X,Y); qb nlD\
% idx = r<=1; w+rw<,u%
% z = nan(size(X)); % /zHL?RqJ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Nj6Np^@sH
% figure h%:wIkZ/
% pcolor(x,x,z), shading interp &>B|?d
% axis square, colorbar Nt+UL/1]
% title('Zernike function Z_5^1(r,\theta)')
UDpI @
% .*k!Zl*
% Example 2: ;$a|4_U$m
% ~l+~MB
% % Display the first 10 Zernike functions F`Vp
% x = -1:0.01:1; yPbOiA*lHz
% [X,Y] = meshgrid(x,x); hQSJt[8My
% [theta,r] = cart2pol(X,Y); \l6mXIn=>
% idx = r<=1; j@Us7Q)A(
% z = nan(size(X)); \@2sI
% n = [0 1 1 2 2 2 3 3 3 3]; I |D]NY^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 7Ph+Vs+h
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8/34{2048
% y = zernfun(n,m,r(idx),theta(idx)); '9wD+'c=A
% figure('Units','normalized') ZG)C#I1;O
% for k = 1:10 (aCl*vV1
% z(idx) = y(:,k); WY~}sE
% subplot(4,7,Nplot(k))
uP ?gGo
% pcolor(x,x,z), shading interp "ZVBn!
% set(gca,'XTick',[],'YTick',[]) ~w%Z Bp
% axis square dR/UXzrc
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fs]Zw mA^
% end $j}sxxTT
% OEGAwP?F
% See also ZERNPOL, ZERNFUN2. a( {`<F
skP_us~
% Paul Fricker 11/13/2006 0\}j[-`pF
CM's6qhQnn
XWy
iS\
% Check and prepare the inputs: kAk,:a;P
% ----------------------------- f&cG;Y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +Vf|YLbhJ
error('zernfun:NMvectors','N and M must be vectors.') yZ)ScB^
end izWl5}+'B
|]\bgh
if length(n)~=length(m) @x J^JcE
error('zernfun:NMlength','N and M must be the same length.') u!`C:C'
end :3n.nKANr
quUJ%F
n = n(:); yRi/YR#
m = m(:); SCH![Amq
if any(mod(n-m,2)) m[7:p{
error('zernfun:NMmultiplesof2', ...
]D-48o0
'All N and M must differ by multiples of 2 (including 0).') tB3CX\e
end Po4cbFZ
7VXeu+-P
if any(m>n) lM1!2d'P
error('zernfun:MlessthanN', ... M>J ADt_]
'Each M must be less than or equal to its corresponding N.') W=\dsdnu*
end eo_T.q
E\*",MGL
if any( r>1 | r<0 ) XZ&v3ul
error('zernfun:Rlessthan1','All R must be between 0 and 1.') &"h!SkX/
end rWs5s!l,
uY~A0I5Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G!;[If:<e
error('zernfun:RTHvector','R and THETA must be vectors.') #"{8Z&Z
end KUyJ"q<W
[;f"',)y,
r = r(:); {|E7N"Qzg
theta = theta(:); u"gp">
length_r = length(r); v+sbRuo8
if length_r~=length(theta) Mv=cLG?X
error('zernfun:RTHlength', ... JrAc]=
'The number of R- and THETA-values must be equal.') 6 {Z\cwP)c
end =x'%zUgE
k >CtWV5B
% Check normalization: ;LE
@Ezx
% -------------------- mD)O\.uA
if nargin==5 && ischar(nflag) WCu%@hh=h
isnorm = strcmpi(nflag,'norm'); }aM`Jp-O
if ~isnorm ;/T-rVND
error('zernfun:normalization','Unrecognized normalization flag.') $SVGpEw
end |u;PU`^-z
else p ri{vveN@
isnorm = false; Gnt!!1_8L
end r&sOM_BUF
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% # :w2Hf6Q
% Compute the Zernike Polynomials .boizW1+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ug"<\"
9/(c cj
% Determine the required powers of r: QS*cd|7J;
% ----------------------------------- A]%t0>EL<
m_abs = abs(m); =>\-ma+
rpowers = []; (x0*(*A}
for j = 1:length(n) ]UT|BE4v
rpowers = [rpowers m_abs(j):2:n(j)]; qU*&49X
end ko2j|*D6@~
rpowers = unique(rpowers); /&as)
M.N~fSJ
% Pre-compute the values of r raised to the required powers, WAXts]=
% and compile them in a matrix: yUmsE-W
% ----------------------------- yL
x .#kx6
if rpowers(1)==0 [RPAkp
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D *I;|.=u
rpowern = cat(2,rpowern{:}); ;GFB@I@
rpowern = [ones(length_r,1) rpowern]; uoY`qF.`
else a<wQzgxG
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =l2Dm
rpowern = cat(2,rpowern{:}); [7Lxt
end k;_KKvQ
14n="-9
% Compute the values of the polynomials: tK|9qs<%
% -------------------------------------- \btR^;_\A
y = zeros(length_r,length(n)); Ig Vo%)n
for j = 1:length(n) w-H%B`/
s = 0:(n(j)-m_abs(j))/2; %:w% o$
pows = n(j):-2:m_abs(j); >[H&k8\7n
for k = length(s):-1:1 Uy59zB2|=
p = (1-2*mod(s(k),2))* ... IFrb}yH
prod(2:(n(j)-s(k)))/ ... :x""E5H
prod(2:s(k))/ ... !q$&JZY
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... qH
h'l;.
prod(2:((n(j)+m_abs(j))/2-s(k))); +eBMn(7Cgv
idx = (pows(k)==rpowers); UGmuX:@y76
y(:,j) = y(:,j) + p*rpowern(:,idx); Dpdn%8+Z
end O| 1f^_S/
t$H':l0
if isnorm sArje(5Eo
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); L`jB)wF/J
end xz"Z3B
end ~[zFQ)([
% END: Compute the Zernike Polynomials vr{'FMc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A-a17}fta
oI_oz0nHk
% Compute the Zernike functions: q}<.x8\
% ------------------------------ qukjS#>+
idx_pos = m>0; :F7k{~
idx_neg = m<0; ~5r=FF6
c Q(}^KO
z = y; K,eqD<
if any(idx_pos) v)@,:u)
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Pgn_9Y?<
end r]k*7PK
if any(idx_neg) `E3:;|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [3j$ 4rP
end y#S1c)vU
N@Xg5huO
% EOF zernfun