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 b.&YUg[#
function z = zernfun(n,m,r,theta,nflag) r MlNp?{_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7(Kc9sJC%%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0b'R5I.M
% and angular frequency M, evaluated at positions (R,THETA) on the ":ycyN@g
% unit circle. N is a vector of positive integers (including 0), and EK_^#b
% M is a vector with the same number of elements as N. Each element J;dFmZOk
% k of M must be a positive integer, with possible values M(k) = -N(k) #4>F%_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, dGe
% and THETA is a vector of angles. R and THETA must have the same ;U&VPIX$
% length. The output Z is a matrix with one column for every (N,M) X*Zv,Wm
% pair, and one row for every (R,THETA) pair. 75f.^4/%
% FReK
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike jYv
!}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @$]h[
% with delta(m,0) the Kronecker delta, is chosen so that the integral D5x^O2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p20Nk$.
% and theta=0 to theta=2*pi) is unity. For the non-normalized |1o]d$3m
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4tjRju?
% p
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% The Zernike functions are an orthogonal basis on the unit circle. 6%-2G@6d
% They are used in disciplines such as astronomy, optics, and Ai;Pht9qi
% optometry to describe functions on a circular domain. 65v'/m!ys
% #A!0KN;GC2
% The following table lists the first 15 Zernike functions. G)Y!aX
% 566EMy|
% n m Zernike function Normalization iKwVYL
% -------------------------------------------------- <3KrhhH
% 0 0 1 1 S%2qB;uw
% 1 1 r * cos(theta) 2 ln5On_Wm
% 1 -1 r * sin(theta) 2 =RA6 p
% 2 -2 r^2 * cos(2*theta) sqrt(6) c1[;a>
% 2 0 (2*r^2 - 1) sqrt(3) gQ~4udla.
% 2 2 r^2 * sin(2*theta) sqrt(6) @p;4g_F
% 3 -3 r^3 * cos(3*theta) sqrt(8) l}x{.q7Ul
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) \] K-<&f
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) UWHC]V?
% 3 3 r^3 * sin(3*theta) sqrt(8) H UjmJu6f{
% 4 -4 r^4 * cos(4*theta) sqrt(10) bHCd|4e,2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W3b\LnUa
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2r,fF<WQ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TR |; /yJ
% 4 4 r^4 * sin(4*theta) sqrt(10) e(Verd:c
% -------------------------------------------------- #qWEyb2UZ
% qF?S[Z;
% Example 1: (_* a4xGF
% dx^3(#B
% % Display the Zernike function Z(n=5,m=1) ;1KhUf;&F
% x = -1:0.01:1; pmC@ fB
% [X,Y] = meshgrid(x,x); /bWV`*
% [theta,r] = cart2pol(X,Y); IX}l)t[:(
% idx = r<=1; E]
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% z = nan(size(X)); 4{|lzo'&
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _R]h]<TQ
% figure xK/`XY
% pcolor(x,x,z), shading interp k(MQ:9'|
% axis square, colorbar +=R:n^r^,
% title('Zernike function Z_5^1(r,\theta)') hRP0Djc
% O1z>A
% Example 2: Xe5J
% bnlL-]]9z
% % Display the first 10 Zernike functions `F)Iv:;y,
% x = -1:0.01:1; IAfYlS#<yD
% [X,Y] = meshgrid(x,x); |:\h3M
% [theta,r] = cart2pol(X,Y); hm&~6rB
% idx = r<=1; .}tL:^'~o
% z = nan(size(X)); Z5\6ca
% n = [0 1 1 2 2 2 3 3 3 3]; "-a>Uj")%
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8)i\d`
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ?mV[TM{p
% y = zernfun(n,m,r(idx),theta(idx)); ]SQ_*$`
% figure('Units','normalized') T/H*Bo*=5
% for k = 1:10 9DIG K\
% z(idx) = y(:,k); r
)T`?y
% subplot(4,7,Nplot(k)) 3yTBkFI!
% pcolor(x,x,z), shading interp {Z|C
% set(gca,'XTick',[],'YTick',[]) ^3el-dZ
% axis square "PX~Yc
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /(q*
% end bc+'n
% 4o%hH
% See also ZERNPOL, ZERNFUN2. 8'zwyd3
@FQ@*XD
% Paul Fricker 11/13/2006 9U+^8,5
2-$R@
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^!A{ 4NV
% Check and prepare the inputs: b&LhydaJ
% ----------------------------- Va1|XQ<CL
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "MyYu}AD
error('zernfun:NMvectors','N and M must be vectors.') 4-m}W;igu
end `aCcTs7~]p
QPBf++|
if length(n)~=length(m) C4b3ZcD2
error('zernfun:NMlength','N and M must be the same length.') 1f}Dza9
end V482V#BP
er 97&5
n = n(:); 0py0zE6,,
m = m(:); Q 5Ln'La$
if any(mod(n-m,2)) 9{+B lNZ
error('zernfun:NMmultiplesof2', ... d@C93VYp
'All N and M must differ by multiples of 2 (including 0).') Z
rvb
%
end ]+J]}C]\d
tf>?;
if any(m>n) aa$+(
error('zernfun:MlessthanN', ... ]Fa VKC~3
'Each M must be less than or equal to its corresponding N.') `LNRl'Zm
end }APf^Ry
u\6:Txqq
if any( r>1 | r<0 ) `TAhW
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .rwZ`MP
end T,k`WR
).k=[@@V
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) lx%<oC+M
error('zernfun:RTHvector','R and THETA must be vectors.') H=B8'N
end [^aow-4z
;X\,-pjv
r = r(:); ni9/7
theta = theta(:); x H\5T!
length_r = length(r); la
f b^
if length_r~=length(theta) e5MX5 T^
error('zernfun:RTHlength', ... mhh8<BI
'The number of R- and THETA-values must be equal.') |',MgA
end Uh*V>HA#
N{f RZN
% Check normalization: mlX^5h'
% -------------------- ,LG6py&aT
if nargin==5 && ischar(nflag) )
_"`{2
isnorm = strcmpi(nflag,'norm'); X5=Dc+
if ~isnorm "(/.3`g
error('zernfun:normalization','Unrecognized normalization flag.') l,L#y4#
end |]^OX$d
else q0$}MB6
isnorm = false; waldLb>7D
end 1)H+iN|im/
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Qv5m^>vj
% Compute the Zernike Polynomials ShFSBD\M#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sFMSH:5z
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% Determine the required powers of r: :4/RB%)"
% ----------------------------------- rD
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m_abs = abs(m); 9tWu>keu
rpowers = []; "\Z.YZUa\
for j = 1:length(n) X&pK#=
rpowers = [rpowers m_abs(j):2:n(j)]; zJOL\J'
end YrFB~z.V
rpowers = unique(rpowers); WM~@/J
89@gYA"Su
% Pre-compute the values of r raised to the required powers, )mS
Aog<
% and compile them in a matrix: #1+1 q{=Z<
% ----------------------------- G)G5eXXX
if rpowers(1)==0 ,)|nxX
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hxGZ}zq*S
rpowern = cat(2,rpowern{:}); ):31!IC
rpowern = [ones(length_r,1) rpowern]; ymiOtA Z
else ilHZx2k
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Rra<MOR
rpowern = cat(2,rpowern{:}); QJjqtOf>
end tY~EB.%
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% Compute the values of the polynomials: s,)Z8H
% -------------------------------------- .k|8nNj
y = zeros(length_r,length(n)); \x5b=~/
for j = 1:length(n) N*gnwrP{
s = 0:(n(j)-m_abs(j))/2; 7='lu;=,
pows = n(j):-2:m_abs(j); 6=0"3%jn@
for k = length(s):-1:1 jTH,GF
p = (1-2*mod(s(k),2))* ... q ^Un,h64t
prod(2:(n(j)-s(k)))/ ... >hQeu1 ~W
prod(2:s(k))/ ... 3dTz$s/[
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Ko|nF-r_
prod(2:((n(j)+m_abs(j))/2-s(k))); 9@/X;zO
idx = (pows(k)==rpowers); O4dJ> O
y(:,j) = y(:,j) + p*rpowern(:,idx);
hRHqG
end ?A+-k4l
b*&AIiT
if isnorm -<h4I
aM
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =dSH8C"
end CB]#`|f
end c@>Tzk%?"
% END: Compute the Zernike Polynomials m-Z<zEQ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dj>zy
3|x*lmit
% Compute the Zernike functions: wc`UcGO
% ------------------------------ xkV(E!O
idx_pos = m>0; x ]{}y_
idx_neg = m<0; Y@B0.5U2
8w/$!9[
z = y; 7uQiP&v
if any(idx_pos) -j9Wf=
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .5*5S[
end c&me=WD
if any(idx_neg) Is57)(^.-
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =z#6mSx|W
end ?gD^K,A Hd
X?whyD)vE@
% EOF zernfun