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 1D3dYVE
function z = zernfun(n,m,r,theta,nflag) tRpL0 =y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j=!(F`/
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z{8exym
% and angular frequency M, evaluated at positions (R,THETA) on the $X{B*
WF
% unit circle. N is a vector of positive integers (including 0), and oP 6.t-<dU
% M is a vector with the same number of elements as N. Each element U4
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% k of M must be a positive integer, with possible values M(k) = -N(k) ^!-E`<jW8
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )Gu0i7iN
% and THETA is a vector of angles. R and THETA must have the same P':]A{<Z
% length. The output Z is a matrix with one column for every (N,M) P 'FPe55F
% pair, and one row for every (R,THETA) pair. Y`E{E|J
% >llwNT
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike S|O%h}AH;
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ySPlyhGF
% with delta(m,0) the Kronecker delta, is chosen so that the integral GgZEg
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, D]LFX/hlH
% and theta=0 to theta=2*pi) is unity. For the non-normalized ~jgN_jz
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. C.Wms}XA
% P22y5z~
% The Zernike functions are an orthogonal basis on the unit circle. cP$wI;P
% They are used in disciplines such as astronomy, optics, and Q0[CH~
% optometry to describe functions on a circular domain. ~{3o(gzl
% 6qmo
ZAg
% The following table lists the first 15 Zernike functions. 5 O{Ip-
% 5Tcl<Y6l
% n m Zernike function Normalization 7>c 0V&
% -------------------------------------------------- l>[QrRXiSN
% 0 0 1 1 )edU <1P
% 1 1 r * cos(theta) 2 )f:!#v(K
% 1 -1 r * sin(theta) 2 6cgpg+-a
% 2 -2 r^2 * cos(2*theta) sqrt(6) `gBXeG2fn
% 2 0 (2*r^2 - 1) sqrt(3) %Hl:nT2M
% 2 2 r^2 * sin(2*theta) sqrt(6) D!OG307P
% 3 -3 r^3 * cos(3*theta) sqrt(8) !)l%EJngL
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) t2!$IHE:
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) +0JH"L5!
% 3 3 r^3 * sin(3*theta) sqrt(8) Rd@n?qB
% 4 -4 r^4 * cos(4*theta) sqrt(10) f"Vm'0r
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?*MV
^IY
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) US*<I2ZLh
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f;_K}23
% 4 4 r^4 * sin(4*theta) sqrt(10) Cs6zv>SR
% -------------------------------------------------- u\Erta`
% CoKj'jA
% Example 1: ?
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% )sz2 9
% % Display the Zernike function Z(n=5,m=1) \CEnOq
% x = -1:0.01:1; v2W"+QS}u
% [X,Y] = meshgrid(x,x); ys"mP*wD
% [theta,r] = cart2pol(X,Y); d
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% idx = r<=1; :n%sU*'T
% z = nan(size(X)); (VF4FC
% z(idx) = zernfun(5,1,r(idx),theta(idx)); y 1jGf83
% figure VgC9'"|
% pcolor(x,x,z), shading interp uq#h\p|
% axis square, colorbar Q e2/4j4
% title('Zernike function Z_5^1(r,\theta)') ?'8MI|*l%
% \qK}(xq[
% Example 2: Zia|`}peW
% b\e)PUm#u@
% % Display the first 10 Zernike functions XQg%*Rw+t
% x = -1:0.01:1; {bq-: CZe
% [X,Y] = meshgrid(x,x); >TJKH^7n
% [theta,r] = cart2pol(X,Y); b6E8ase:F
% idx = r<=1; X0r#,u
% z = nan(size(X)); ~%!U,)-
% n = [0 1 1 2 2 2 3 3 3 3]; =LeVJGF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; aR(Z~z;C
% Nplot = [4 10 12 16 18 20 22 24 26 28]; #t9=qR~"
% y = zernfun(n,m,r(idx),theta(idx)); K:mL%o2J
% figure('Units','normalized') }FdcbNsP
% for k = 1:10 D*2p
% z(idx) = y(:,k); LZAj4|~,m
% subplot(4,7,Nplot(k)) 77bZ
% pcolor(x,x,z), shading interp NtP.)
% set(gca,'XTick',[],'YTick',[]) Y_ ;i
% axis square ^zluO
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) YC,.Y{oY{
% end #+DmH
% JI#Enh!Lv
% See also ZERNPOL, ZERNFUN2. _F$t#.o
Nz;*;BQK:
% Paul Fricker 11/13/2006 @xM!:
PAWr1]DI
#o |&MV_j
% Check and prepare the inputs: QIz N#;g
% ----------------------------- hZ /
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f8_UIdM7
error('zernfun:NMvectors','N and M must be vectors.') z%gtV'
end -D^y)
nJ0eZBgB]
if length(n)~=length(m) `/j|Rb|eow
error('zernfun:NMlength','N and M must be the same length.') ~esEql=Q3'
end {O,M}0Eg
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n = n(:); r D!.N
m = m(:); nm|m1Z+U
if any(mod(n-m,2)) t=\[J+
error('zernfun:NMmultiplesof2', ... CR PE?CRQF
'All N and M must differ by multiples of 2 (including 0).') vz_g2.7l\
end YKxA2`3v%
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if any(m>n) L-:L=
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error('zernfun:MlessthanN', ... oHFDg?Z`
'Each M must be less than or equal to its corresponding N.') 2bG4,M
end JhXN8Bq33
yt#;3
if any( r>1 | r<0 ) =4\~M"[p
error('zernfun:Rlessthan1','All R must be between 0 and 1.') >nW}zkfn
end c]v3dHE_h
GyM%vGl
3
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5i-;bLm
error('zernfun:RTHvector','R and THETA must be vectors.')
o*ED!y7
end |DS@90}
cb&In<q
r = r(:); bRe *(
theta = theta(:); _eeX]xSSl
length_r = length(r); Pisr&"A
if length_r~=length(theta) ?D 9#dGK
error('zernfun:RTHlength', ... W%ZU& YBc
'The number of R- and THETA-values must be equal.') ;Sl0kSu
end ]~eWr2uG?
mSw?iL
% Check normalization: bc}OmPE
% -------------------- 'Mhdw}
if nargin==5 && ischar(nflag) V~"d`j
isnorm = strcmpi(nflag,'norm'); U$J_:~
if ~isnorm &fhurzzAm
error('zernfun:normalization','Unrecognized normalization flag.') Bo(l !G
end 8VGXw;(Y,d
else
]p.f*]
isnorm = false; ,$ret@.H
end {+mkXp])R
L"<Eov6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /1
%0A
% Compute the Zernike Polynomials -t#a*?"$w
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aq| [g
vX24W*7
% Determine the required powers of r: #/=yz<B
% ----------------------------------- s(LqhF[N2]
m_abs = abs(m); fB}5,22
rpowers = []; d"a7{~l
for j = 1:length(n) PY<V
rpowers = [rpowers m_abs(j):2:n(j)]; 717m.t,x
end <?}g[]i
rpowers = unique(rpowers); hRcJ):Wyb
9+|,aG s
% Pre-compute the values of r raised to the required powers, 2Yjysn
% and compile them in a matrix: +6-!o,(
% ----------------------------- =W^L8!BE'
if rpowers(1)==0 )O(Gw-jWE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Nn\\}R
rpowern = cat(2,rpowern{:}); xF31%b`z:
rpowern = [ones(length_r,1) rpowern]; Ci:QIsu*
else -^"?a]B
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :m)?+
rpowern = cat(2,rpowern{:}); ]}c=U@D,9
end }=4".V`-o
f#MN-1[67
% Compute the values of the polynomials: +'4 dP#
% -------------------------------------- Db:WAjU
y = zeros(length_r,length(n)); tC~itU=V
for j = 1:length(n) {<BK@U
s = 0:(n(j)-m_abs(j))/2; |?W
pows = n(j):-2:m_abs(j); [=!MS?-G
for k = length(s):-1:1 l'f!za0
p = (1-2*mod(s(k),2))* ... py4_hj\v
prod(2:(n(j)-s(k)))/ ... tTamFL6
prod(2:s(k))/ ... ]gk1h=Y~h
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... QX|K(`of
prod(2:((n(j)+m_abs(j))/2-s(k))); @SB+u+mOS
idx = (pows(k)==rpowers); DZZt%n8J
y(:,j) = y(:,j) + p*rpowern(:,idx); )j*qGsOg
end ,Ou)F;r
cv1L!Ce,
if isnorm je%12DM
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); g"f^YEQ_
end Inoou'jX
end yh<aFYdk
% END: Compute the Zernike Polynomials I{bi3y0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xb>+~5 9:
N `MQHQ1
% Compute the Zernike functions: ~`.%n7
% ------------------------------ J n/=v\K@
idx_pos = m>0; \}W.RQ^3
idx_neg = m<0; $
7!GA9Bn
#1k,t
z = y; T]`"
Xl8
if any(idx_pos) WLb7]rCTp
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Op~+yMef
end H0 t1& :
if any(idx_neg) u>Hx#R<*%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); AR^Di`n!
end [8#l~
|U
&9tsk#bA.g
% EOF zernfun