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 vx:MLmZ.
function z = zernfun(n,m,r,theta,nflag) $$UMc-Pq
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7MRu=Z.-b
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'S_kD! BO
% and angular frequency M, evaluated at positions (R,THETA) on the XCQS_'D
% unit circle. N is a vector of positive integers (including 0), and ~]+-<O^U~
% M is a vector with the same number of elements as N. Each element u/`jb2eEU:
% k of M must be a positive integer, with possible values M(k) = -N(k) @x9DV{j)V
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, N|Cx";,|FZ
% and THETA is a vector of angles. R and THETA must have the same Kk5 vC{
% length. The output Z is a matrix with one column for every (N,M) W<J".2D
% pair, and one row for every (R,THETA) pair. W/z\j/Rgc
% *?;<buJb?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ix+===6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RYU(z;+0p
% with delta(m,0) the Kronecker delta, is chosen so that the integral ~Wh}W((L
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eY3l^Su1
% and theta=0 to theta=2*pi) is unity. For the non-normalized kOv2E]
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5hNjJqu
% K\Oz
~,z
% The Zernike functions are an orthogonal basis on the unit circle. 4vri=P 2%
% They are used in disciplines such as astronomy, optics, and h'{}eYb+
% optometry to describe functions on a circular domain. 5F@7A2ZR
% 9fk@C /$
% The following table lists the first 15 Zernike functions. VieX5
% |K},f,
% n m Zernike function Normalization czMu<@c [
% -------------------------------------------------- #+mt}w/
% 0 0 1 1 6pkZ8Vp:
% 1 1 r * cos(theta) 2 %s.hqr,I
% 1 -1 r * sin(theta) 2 fz%I'+!
% 2 -2 r^2 * cos(2*theta) sqrt(6) "AN2K
% 2 0 (2*r^2 - 1) sqrt(3) =[wVRQ?
% 2 2 r^2 * sin(2*theta) sqrt(6) ;]ojfR=?%
% 3 -3 r^3 * cos(3*theta) sqrt(8) %O5
k+~9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) dXAKk[uf
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) "CYh"4]@rD
% 3 3 r^3 * sin(3*theta) sqrt(8) v 4@=>L
% 4 -4 r^4 * cos(4*theta) sqrt(10) :D-xa!7
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nC^|83
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2o0.ttBAqZ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f/spJ<B).4
% 4 4 r^4 * sin(4*theta) sqrt(10) (kpn"]^'
% -------------------------------------------------- ML6V,V/e
% 7X3<8:%
% Example 1: }-3|
v<d
% ;#np~gL
% % Display the Zernike function Z(n=5,m=1) &!I^m
% x = -1:0.01:1; Evd>s
% [X,Y] = meshgrid(x,x); Da#|}m0>
% [theta,r] = cart2pol(X,Y); 1}#(4tw)
% idx = r<=1; *9"L?S(X#
% z = nan(size(X)); 19)fN-0Z
% z(idx) = zernfun(5,1,r(idx),theta(idx)); [al, UO
% figure d*%-r2K
% pcolor(x,x,z), shading interp Am~ NBQ7
% axis square, colorbar fH_G;#q
% title('Zernike function Z_5^1(r,\theta)') M8Y\1#~
% \cq
gCab/2
% Example 2: B_FfXFQm<
% @Q:5{?
% % Display the first 10 Zernike functions ,E]u[7A
% x = -1:0.01:1; %|(~k*s4
% [X,Y] = meshgrid(x,x); PV?XpT
% [theta,r] = cart2pol(X,Y); 0sjw`<ic
% idx = r<=1; pg3B^
% z = nan(size(X)); 9*!C|gC9Ia
% n = [0 1 1 2 2 2 3 3 3 3]; 8l|v#^v
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; )A]E:]2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; "hRw_<
% y = zernfun(n,m,r(idx),theta(idx)); zx7*Bnu0
% figure('Units','normalized') {7^7)^@
% for k = 1:10 .e2qa
% z(idx) = y(:,k); ?#@JH
% subplot(4,7,Nplot(k)) H-%)r&"vn
% pcolor(x,x,z), shading interp *&X.
% set(gca,'XTick',[],'YTick',[]) &gc8"B@V
% axis square ajy.K'B*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) uMm/$#E
% end '>:mEXK}w
% }{*((@GY}
% See also ZERNPOL, ZERNFUN2. /p~Wk4'
Qh%(yL!
% Paul Fricker 11/13/2006 ]JQk,<l5E
[3`T/Wm
1nh2()QI[
% Check and prepare the inputs: tN|sHgs
% ----------------------------- G!~[+B
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L2>UA<@mZ
error('zernfun:NMvectors','N and M must be vectors.') q|J3]F !n
end jREj]V>
\M>+6m@w
if length(n)~=length(m) pyK|zvr-r
error('zernfun:NMlength','N and M must be the same length.') s MAc+9G9k
end >j1\]uo
'>(R'g42n
n = n(:); 84[T!cDk
m = m(:); eWO^n>Y
if any(mod(n-m,2)) mLM$dk3
error('zernfun:NMmultiplesof2', ... L{$ZL &
'All N and M must differ by multiples of 2 (including 0).') ^.Y"<oZSS
end o"@y=n/
2BOe,giy
if any(m>n) 't=\YFQ*v
error('zernfun:MlessthanN', ... ADRjCk}I
'Each M must be less than or equal to its corresponding N.') =p>"PqJ/7n
end ~o`I[-g)
q#B^yk|Y
if any( r>1 | r<0 ) nf!RB-orF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4cK6B)X
end qPdNI1 |
b7>^w<ki
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) '>[KVvm
error('zernfun:RTHvector','R and THETA must be vectors.') {Q8DPkW
end iZ+\vO?|
bL5z%bV
r = r(:); 2A@9jl s
theta = theta(:); XtfO;`
length_r = length(r); }*l V
if length_r~=length(theta) TEOV>Tt
error('zernfun:RTHlength', ... W#|]m=2W
'The number of R- and THETA-values must be equal.') N1WP
end ?iG}Qj@5
?}%Gr,tj2
% Check normalization: FQ?,&s$Bmd
% -------------------- z<rdxn,9
if nargin==5 && ischar(nflag) V#!ihL/>
isnorm = strcmpi(nflag,'norm'); HGmgQ>q@M$
if ~isnorm 9z5K -s
error('zernfun:normalization','Unrecognized normalization flag.') ws5x53K
end J=67As
else /_E:sI9(
isnorm = false; 0B)l"$W[)/
end f&t]O$
VtF^;
f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xI'<4lo7Z
% Compute the Zernike Polynomials >%+"-bY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dz.]5R
]@1YgV
% Determine the required powers of r: DR/qe0D
% ----------------------------------- ?_ [xpK()
m_abs = abs(m); o#E 3{zM
rpowers = []; Ea1{9>S
for j = 1:length(n) =nOV!!
rpowers = [rpowers m_abs(j):2:n(j)]; HyXw^ +tsj
end EDvK9J
rpowers = unique(rpowers); tA$,4B?
~6@zXHAS
% Pre-compute the values of r raised to the required powers, 8 f%@:}H
% and compile them in a matrix: {
yU1db^
% ----------------------------- I})la!9
if rpowers(1)==0 _:0<]<x?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *=dFTd"#
rpowern = cat(2,rpowern{:}); 4NbX!"0
rpowern = [ones(length_r,1) rpowern]; )eGGA6G
else bv0B
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n1o/-UY
rpowern = cat(2,rpowern{:}); CmRn
end AL! ^1hCF
y4)M,+O5
% Compute the values of the polynomials: g^8dDY[%
% -------------------------------------- ,Ihuo5>/z
y = zeros(length_r,length(n)); Pca~V>Hd
for j = 1:length(n) pOD|
s = 0:(n(j)-m_abs(j))/2; 8-cG[/|0
pows = n(j):-2:m_abs(j); "e g`3v
for k = length(s):-1:1 !`\W8JT+
p = (1-2*mod(s(k),2))* ... ^G=wRtS
prod(2:(n(j)-s(k)))/ ... 'T7JXV5
prod(2:s(k))/ ... Gk,{{:M:5
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jpyV52
prod(2:((n(j)+m_abs(j))/2-s(k))); WM: ~P$%cx
idx = (pows(k)==rpowers); _`/0/69
y(:,j) = y(:,j) + p*rpowern(:,idx); 5. :To2
end JWy$` "{
tu77Sb
if isnorm Nv*x^y]
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :q~qRRmjBe
end SDiZOypS
end _baqN!N
% END: Compute the Zernike Polynomials |`s}PcV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B+);y
= Ii@-C
% Compute the Zernike functions: swG^L$r`
% ------------------------------ cGkl=-oQ'
idx_pos = m>0; riZFcVsB
idx_neg = m<0; 0ang~_
' F`*(\#
z = y; g}Hk4+
if any(idx_pos) jp8=>mk
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Sn.I
]:l
end #"ayq,GC<
if any(idx_neg) vKAHf;1
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oHI~-{m3)
end 2P$l XGjh
r {)d?Ho=
% EOF zernfun