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 9h/Hy aN
function z = zernfun(n,m,r,theta,nflag) 3m$ck$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. tSe[*V4{'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $z`l{F4eMf
% and angular frequency M, evaluated at positions (R,THETA) on the C-\3,
% unit circle. N is a vector of positive integers (including 0), and !#zO%
% M is a vector with the same number of elements as N. Each element > `mV^QD
% k of M must be a positive integer, with possible values M(k) = -N(k) h^
K]ASj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Ahc9HA2
% and THETA is a vector of angles. R and THETA must have the same +,cd$,18
% length. The output Z is a matrix with one column for every (N,M) 6AoKuT;
% pair, and one row for every (R,THETA) pair. X` J86G )
% 34Cnbtq^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j#xGB]
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FmhAUe
% with delta(m,0) the Kronecker delta, is chosen so that the integral $ w+.-Tr
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @1xIph<z
% and theta=0 to theta=2*pi) is unity. For the non-normalized `F]
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }1%%`
% e^,IZ{
% The Zernike functions are an orthogonal basis on the unit circle. `sDLxgwI
% They are used in disciplines such as astronomy, optics, and =dsEt\
j
% optometry to describe functions on a circular domain. $N Mu
% F`GXho[
% The following table lists the first 15 Zernike functions. )%PMDG|
% @|5B}%!
% n m Zernike function Normalization 1xu~@v60
% -------------------------------------------------- #SG.`J<%
% 0 0 1 1 Y`(~eNX^%
% 1 1 r * cos(theta) 2 "0,FB4L[U5
% 1 -1 r * sin(theta) 2 -+M360
% 2 -2 r^2 * cos(2*theta) sqrt(6) (#Xs\IEV F
% 2 0 (2*r^2 - 1) sqrt(3) IRueq @4
% 2 2 r^2 * sin(2*theta) sqrt(6) 7XLqP
% 3 -3 r^3 * cos(3*theta) sqrt(8) gVe]?Jva`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) !
,{zDMA
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) J_fs}Y1q\
% 3 3 r^3 * sin(3*theta) sqrt(8) (z8;J>7
% 4 -4 r^4 * cos(4*theta) sqrt(10) JU.!<
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^d@2Y0hH
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !v(^wqna\
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dwQ1~
% 4 4 r^4 * sin(4*theta) sqrt(10) <*WGvCh%w
% -------------------------------------------------- KVh#"]<WV
% 9X,iQ
% Example 1: KVr9kcs
% |a
a\t
% % Display the Zernike function Z(n=5,m=1) i7Cuc+j8
% x = -1:0.01:1; T?QW$cU!e:
% [X,Y] = meshgrid(x,x); ,RM8D)m\
% [theta,r] = cart2pol(X,Y); ];"40 /X
% idx = r<=1; .ZV='i()X
% z = nan(size(X)); \#WWJh"W
% z(idx) = zernfun(5,1,r(idx),theta(idx)); em5~4;&'
% figure (wu ciKQ
% pcolor(x,x,z), shading interp 5!cp^[rGL
% axis square, colorbar y:^o._
% title('Zernike function Z_5^1(r,\theta)') r>7+&s*yk
% %l14K_
% Example 2: *^Ges;5$"
% ,Q3OQ[Nmh
% % Display the first 10 Zernike functions 97$Q?a8S@
% x = -1:0.01:1; 8|<</v8i
% [X,Y] = meshgrid(x,x); .@%L8_sMR
% [theta,r] = cart2pol(X,Y); _x1W\#
% idx = r<=1; =.&8ghJ*M
% z = nan(size(X)); ?QzL#iO}h
% n = [0 1 1 2 2 2 3 3 3 3]; $v[mIR
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Dr(2@0P
% Nplot = [4 10 12 16 18 20 22 24 26 28]; &M@c50&%
% y = zernfun(n,m,r(idx),theta(idx)); WJu(,zM?G
% figure('Units','normalized') ;6D3>Lm
% for k = 1:10 9<&M~(dwT4
% z(idx) = y(:,k); 9(OeH7
% subplot(4,7,Nplot(k)) 'S9o!hb'@
% pcolor(x,x,z), shading interp E?czolNl
% set(gca,'XTick',[],'YTick',[]) eY'nS
% axis square Yj*T'<e
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) </D.}ia
% end sNcU>qjj6
% IW&*3I<K
% See also ZERNPOL, ZERNFUN2. (,jsZ!sl
m;\nMdn
% Paul Fricker 11/13/2006 WW{_D
o $W@@aM
4w=v
/WDo
% Check and prepare the inputs: F6111Q </
% ----------------------------- 8a`3eM~?[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZOcpF1y
error('zernfun:NMvectors','N and M must be vectors.') yYYP;N?g4k
end WeaT42*Q{
9#:fQ!3`
if length(n)~=length(m) nW"O+s3
error('zernfun:NMlength','N and M must be the same length.') OylUuYy~j
end )^AZmUYZ
HcJ!(
n = n(:); 2uN3:_w
m = m(:); A[^#8evaK
if any(mod(n-m,2)) wK7w[Xt
error('zernfun:NMmultiplesof2', ... XHj%U
'All N and M must differ by multiples of 2 (including 0).') s>I]_W)Pt
end ^)Awjj9
U5wh( vi
if any(m>n) }2LWDQ;po
error('zernfun:MlessthanN', ... gaz",kK<
'Each M must be less than or equal to its corresponding N.') #::+# G
end UkpTK8>&
.\T!oSb4[
if any( r>1 | r<0 ) 7gN;9pc$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ][tR=Y#&y5
end F~fBr
ui,!_O .c
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) b@8z+,_
error('zernfun:RTHvector','R and THETA must be vectors.') %(p9AE
end "{qnm+G
~cSXBc,+
r = r(:); VgIk '.
theta = theta(:); B
}euIQB
length_r = length(r); /CO=!*7fz
if length_r~=length(theta) JxwKTFU'3O
error('zernfun:RTHlength', ... fX 1%I
'The number of R- and THETA-values must be equal.') O50<h O]l
end , +J)`+pJx
IB|6\uKn
% Check normalization: 4gC(zJ
% -------------------- }Vob)r{R@
if nargin==5 && ischar(nflag) f~\H|E8(
isnorm = strcmpi(nflag,'norm'); LEPTL#WT1
if ~isnorm ><D2of|
error('zernfun:normalization','Unrecognized normalization flag.') =E]tEi
end tt2
S.j
else 7F0J*M
isnorm = false; 0Zwx3[bq6K
end /eH37H
HM0&%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }(!Uq
% Compute the Zernike Polynomials (|Gwg \r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `<d.I%}
iYvzZ7
8f
% Determine the required powers of r: ,M?8s2?
% ----------------------------------- 9uWg4U
m_abs = abs(m); ~mt{j7
rpowers = []; (>A#|N1U
for j = 1:length(n) [ !#Dba#
rpowers = [rpowers m_abs(j):2:n(j)]; u28$V]
end PkyX,mr#1
rpowers = unique(rpowers); ~Yg)8
9#P~cW?
% Pre-compute the values of r raised to the required powers, >'q]ypA1
% and compile them in a matrix: ?2da6v,t
% ----------------------------- R|8L'H+1x
if rpowers(1)==0 ~K #92
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *9r(lmrfj
rpowern = cat(2,rpowern{:}); Uv>e :U7 ;
rpowern = [ones(length_r,1) rpowern]; us?q^>u
else
0LL65[
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); mxF+Fp~
rpowern = cat(2,rpowern{:}); 2IW!EUR
end +C7E]0!r
DFQ`(1Q
% Compute the values of the polynomials: Q njK<}M9
% -------------------------------------- ^j${#Q
y = zeros(length_r,length(n)); ibZ[U p?
for j = 1:length(n) WO9vOS>
s = 0:(n(j)-m_abs(j))/2; q?mpvpLG
pows = n(j):-2:m_abs(j); fi>.X99(G
for k = length(s):-1:1 :Ob^b3<t
p = (1-2*mod(s(k),2))* ... .wq
j
prod(2:(n(j)-s(k)))/ ... B,_K mHItd
prod(2:s(k))/ ... 9-{ +U,3)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .hxFFk%5
prod(2:((n(j)+m_abs(j))/2-s(k))); VT4>6u}
idx = (pows(k)==rpowers); H.XyNtJ
y(:,j) = y(:,j) + p*rpowern(:,idx); }]dzY(
end k"gm;,`
hy;V~J#
if isnorm eDP&W$s#
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +U
J~/XV
end xLFMC?I
end u? >x
% END: Compute the Zernike Polynomials ~E8/m_> rU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R'tvF$3=i
.!L{yU,
% Compute the Zernike functions: !9HWx_,|Z
% ------------------------------ w@R" g%k-
idx_pos = m>0;
Nb3O>&J
idx_neg = m<0; *a\x!c"
~a2|W|?
z = y; b49h @G
if any(idx_pos) 8r"-3<*
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @8m%*pBg
end .YvIVQ
if any(idx_neg) ewn\'RLZ"@
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~"\v(\P e
end @|"K"j#
P(I%9
% EOF zernfun