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 L\YKdUL
function z = zernfun(n,m,r,theta,nflag) `mzb(bE
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Rs#|JWB2V
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;hwzYXWF
% and angular frequency M, evaluated at positions (R,THETA) on the bni)Qw
% unit circle. N is a vector of positive integers (including 0), and <FUon
% M is a vector with the same number of elements as N. Each element iU5P$7.p
% k of M must be a positive integer, with possible values M(k) = -N(k) }taLk@T
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ocF>LR%P
% and THETA is a vector of angles. R and THETA must have the same IU|kNBo
% length. The output Z is a matrix with one column for every (N,M) O~ 27/
% pair, and one row for every (R,THETA) pair. G}VDEC
% `?|Rc
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :\b|dvI<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rfs (#
% with delta(m,0) the Kronecker delta, is chosen so that the integral :?=Q39O9
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |O-`5_z$r
% and theta=0 to theta=2*pi) is unity. For the non-normalized o'Wz*oY))\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7X}TB\N1
% BFWi(58q
% The Zernike functions are an orthogonal basis on the unit circle. wiJRCH
% They are used in disciplines such as astronomy, optics, and Vr/Bu4V"
% optometry to describe functions on a circular domain. _({@B`N}
% ZQAO"huk]
% The following table lists the first 15 Zernike functions.
R_1qn
% H_w%'v &
% n m Zernike function Normalization <~{du ?4n
% -------------------------------------------------- SO;N~D1Z6
% 0 0 1 1 :"QfF@Z{
% 1 1 r * cos(theta) 2 *0y{ ~@
% 1 -1 r * sin(theta) 2 S8" f]5s
% 2 -2 r^2 * cos(2*theta) sqrt(6) ~~nqU pK?v
% 2 0 (2*r^2 - 1) sqrt(3) nBz`q+V
% 2 2 r^2 * sin(2*theta) sqrt(6) 0C$8g
Y*
% 3 -3 r^3 * cos(3*theta) sqrt(8)
l{$[}<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $.rzc]s
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #DFp[\)1
% 3 3 r^3 * sin(3*theta) sqrt(8) ~$<UE}qp
% 4 -4 r^4 * cos(4*theta) sqrt(10) I [0!SIqY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rp's
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "AC^ rz~U
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m%QSapV
% 4 4 r^4 * sin(4*theta) sqrt(10) <*(^{a.O
% -------------------------------------------------- ])w[
% T95t"g?p
% Example 1: lpgd#vr
% G.\l qYrXU
% % Display the Zernike function Z(n=5,m=1) hmC*^"C>U=
% x = -1:0.01:1; =\};it{u
% [X,Y] = meshgrid(x,x); ?9mkRd}c
% [theta,r] = cart2pol(X,Y); kn"q:aD
% idx = r<=1; !eI2r
% z = nan(size(X)); f>polxB%N
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7
oQ[FdRn*
% figure a.2L*>p
% pcolor(x,x,z), shading interp S($Su7g%_
% axis square, colorbar mr:CuqJ
% title('Zernike function Z_5^1(r,\theta)') Jr;jRe`4c
% J00VTb`
% Example 2: i-"
p)2d=#
% x,n,Qlb
% % Display the first 10 Zernike functions BU|#e5
% x = -1:0.01:1; CGbwmPx
% [X,Y] = meshgrid(x,x); 3g~'5Ao
% [theta,r] = cart2pol(X,Y); LR(-<"
% idx = r<=1; E"~2./+rd
% z = nan(size(X)); #,d I$gY
% n = [0 1 1 2 2 2 3 3 3 3]; =u[k1s?
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; KNLnn;l
% Nplot = [4 10 12 16 18 20 22 24 26 28]; eE
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% y = zernfun(n,m,r(idx),theta(idx)); X;oa[!k
% figure('Units','normalized') {)8>jxQN
% for k = 1:10 V)(R]BK{
% z(idx) = y(:,k); FRu]kZv2
% subplot(4,7,Nplot(k)) r SkUSe6
% pcolor(x,x,z), shading interp kF"@Ngv.
% set(gca,'XTick',[],'YTick',[]) _Q[$CcDEE
% axis square Gh.[dF?
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @.Icz
% end 9h4({EE2t
% h:Gu`+D>W
% See also ZERNPOL, ZERNFUN2. ).^}AFta
5,-U.B}
% Paul Fricker 11/13/2006 ",7Q
%h?x!,q
Y
PYbVy<xc
% Check and prepare the inputs: 0j"8@<
% ----------------------------- }XO K,Hw
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ez|oN,
error('zernfun:NMvectors','N and M must be vectors.') Ms~{9?
end (8.Z..PH
Sz- Jy:j
if length(n)~=length(m) ( +pLA"xq
error('zernfun:NMlength','N and M must be the same length.') NS%WeAf
end ?qCK7$j
Dn&D!B
n = n(:); &e6CJ
m = m(:); g35DV6
if any(mod(n-m,2)) M`rl!Ci#
error('zernfun:NMmultiplesof2', ... %?e& WLS
'All N and M must differ by multiples of 2 (including 0).') \b%kf9 9
end fFb_J`'ue
]gYz
4OT
if any(m>n) CC#;c1t
error('zernfun:MlessthanN', ... ^*P%=>zO
'Each M must be less than or equal to its corresponding N.') N"nd*?
end o.0ci+z@
ZI= %JU(
if any( r>1 | r<0 ) *h}XWB C1q
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6d_'4B
end O3I8k\`
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ZosP(Tdq
error('zernfun:RTHvector','R and THETA must be vectors.') E\Rhz]G(
end =EHUR'
W[Ls|<Q
r = r(:); &*+'>UEe5
theta = theta(:); O^oWG&Y;v
length_r = length(r);
TWA-.>c
if length_r~=length(theta) V5UF3'3;}
error('zernfun:RTHlength', ... _f$^%?^
'The number of R- and THETA-values must be equal.') _d5QbTe
end i\,-oO
N@t|7~
% Check normalization: etTn_v
% -------------------- u6AA4(
if nargin==5 && ischar(nflag) -[cTx[Z,
isnorm = strcmpi(nflag,'norm'); Qk:Y2mL
if ~isnorm o,_?^'@
error('zernfun:normalization','Unrecognized normalization flag.') e
9;~P}
end gt@m?w(
else uG,5BV .M
isnorm = false; f|\onHI)>
end f&Gt|
3kybLOG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vSEuk}pk
% Compute the Zernike Polynomials 17%Mw@+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aDU<wxnSvO
sB7#
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% Determine the required powers of r: h1de[q)
% ----------------------------------- 9Z4nAc
m_abs = abs(m); >T^;MS
rpowers = []; Fld=5B^}
for j = 1:length(n) 6 (]Dh;gC
rpowers = [rpowers m_abs(j):2:n(j)]; A^USBv+9`
end `sn^ysp
rpowers = unique(rpowers); '=b/6@&
5IE#\FITO|
% Pre-compute the values of r raised to the required powers, Ayxkv)%:@)
% and compile them in a matrix: nT7%j{e=L
% ----------------------------- y
[}.yyye
if rpowers(1)==0 H?yK~bGQ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -|$@-fY;
rpowern = cat(2,rpowern{:}); !>FYK}c7
rpowern = [ones(length_r,1) rpowern]; (A9Fhun
else J')o|5S1N
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ag [ZW
rpowern = cat(2,rpowern{:}); ?9
<:QE;I>
end (ZUHvvL
-r`.#c4
% Compute the values of the polynomials: gb[5&>(#
% -------------------------------------- 6m}Ev95
y = zeros(length_r,length(n)); {$0mwAOH "
for j = 1:length(n) Ag-(5:
s = 0:(n(j)-m_abs(j))/2; igCZ|Ru\
pows = n(j):-2:m_abs(j); fDv2JdiU
for k = length(s):-1:1 <FV1Wz
p = (1-2*mod(s(k),2))* ... .s?L^Z^
prod(2:(n(j)-s(k)))/ ... &*M!lxDN
prod(2:s(k))/ ...
dm\F
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]E5o1eeg
prod(2:((n(j)+m_abs(j))/2-s(k))); }|h# \$w
idx = (pows(k)==rpowers); KLST\Ln:
y(:,j) = y(:,j) + p*rpowern(:,idx); YL!P0o13r
end (nQ^
xG~P+n7t5$
if isnorm l!D}3jD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5'OrHk;u
end c[0}AGJ
end qU \w=
% END: Compute the Zernike Polynomials q}3`|'3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x[
SDl(<@;
(~p<
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% Compute the Zernike functions: R$R *'l
% ------------------------------ IPS4C[v
idx_pos = m>0; G<L;4nA)
idx_neg = m<0; {5Q!Y&N.%
S,88*F(<^q
z = y; ?qb}?&1
if any(idx_pos) P\E<9*V
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Yj&F;_~
end u+9hL4
if any(idx_neg) ahusta
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ki;*u_4{
end O%\*@4zM
NDN7[7E
% EOF zernfun