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 '<1Q;3Ho
function z = zernfun(n,m,r,theta,nflag) GsiT!OP]y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. o+g\\5s
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /NUu^ N
% and angular frequency M, evaluated at positions (R,THETA) on the <(_${zR
% unit circle. N is a vector of positive integers (including 0), and bo[[<j!"I
% M is a vector with the same number of elements as N. Each element .6A{
% k of M must be a positive integer, with possible values M(k) = -N(k) #B@*-
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pGP$2
% and THETA is a vector of angles. R and THETA must have the same j"9Zaq_
% length. The output Z is a matrix with one column for every (N,M) 5"z~BE7
% pair, and one row for every (R,THETA) pair. xcX^L84\
% DAQozhP8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,
%A2wV
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), J5SOPG
% with delta(m,0) the Kronecker delta, is chosen so that the integral 6.6x$y3v
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,lQfsntk'
% and theta=0 to theta=2*pi) is unity. For the non-normalized J~lKN
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. v-XB\|f
% e_dsBmTh
% The Zernike functions are an orthogonal basis on the unit circle. cdTG ]n
% They are used in disciplines such as astronomy, optics, and r<pt_Cd
% optometry to describe functions on a circular domain. q(Zu;ecBN
% 9&Ny;oy#6
% The following table lists the first 15 Zernike functions. NT<}-^
% 3yu,qb'"&
% n m Zernike function Normalization @!::_E+F]
% -------------------------------------------------- ~QU\kZ7Z
% 0 0 1 1 Bi|-KS.9
% 1 1 r * cos(theta) 2 ZmZ7E]c
% 1 -1 r * sin(theta) 2 oB%j3aAH
% 2 -2 r^2 * cos(2*theta) sqrt(6) qhOV>j,d
% 2 0 (2*r^2 - 1) sqrt(3) =' &TqiIv"
% 2 2 r^2 * sin(2*theta) sqrt(6) Z[9f8/6<b
% 3 -3 r^3 * cos(3*theta) sqrt(8) H;Gd
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) hEAP,)>F
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Xagz(tm/
% 3 3 r^3 * sin(3*theta) sqrt(8) Rip[
% 4 -4 r^4 * cos(4*theta) sqrt(10) Eg0qY\'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D`NQEt"(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z6'l" D'h
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #.|MV}6rQ
% 4 4 r^4 * sin(4*theta) sqrt(10) ]Ab$IKY
% -------------------------------------------------- +?uZ~VSl
% `.YMbj#T
% Example 1: .2/W.z2
% 9On(b|mT
% % Display the Zernike function Z(n=5,m=1) GtkZ%<KF9
% x = -1:0.01:1; J#Agk^Y 5
% [X,Y] = meshgrid(x,x); T9]:,
z
% [theta,r] = cart2pol(X,Y); !N\i9w}
% idx = r<=1; _}Ec[c
% z = nan(size(X)); HfA@tZ5q|U
% z(idx) = zernfun(5,1,r(idx),theta(idx)); s3oQ( wC %
% figure %-3wR@
% pcolor(x,x,z), shading interp z5 :53,`D'
% axis square, colorbar XZ_vbYTj
% title('Zernike function Z_5^1(r,\theta)') Te@=8-u-
% ;{ESo?$*
% Example 2: 7-[^0qS
% qrY]tb^K
% % Display the first 10 Zernike functions $GX9-^og=T
% x = -1:0.01:1; |=#uzp7*
% [X,Y] = meshgrid(x,x); ,{g B$8z^
% [theta,r] = cart2pol(X,Y); *"sDsXo- I
% idx = r<=1; p"o_0{8
% z = nan(size(X)); C%;J9(r
% n = [0 1 1 2 2 2 3 3 3 3]; cfUG)-]P~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Tw!_=zy(Gw
% Nplot = [4 10 12 16 18 20 22 24 26 28]; HsAKz]Mq
% y = zernfun(n,m,r(idx),theta(idx)); EALgBv>#ZL
% figure('Units','normalized') +t<'{KZ7;
% for k = 1:10 u;=a=>05IR
% z(idx) = y(:,k); t"FB}%G
% subplot(4,7,Nplot(k)) at5=Zo[bP
% pcolor(x,x,z), shading interp uOQl;}Lk5
% set(gca,'XTick',[],'YTick',[]) NZt
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% axis square @1+({u#B
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .{66q#.
% end 1n EW'F
% rPF2IS(5
% See also ZERNPOL, ZERNFUN2. /PgcW
PVX23y;
% Paul Fricker 11/13/2006 >kG: MJj
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% Check and prepare the inputs: :DR}lOi`
% ----------------------------- Oo8"s+G
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #~:@H&f790
error('zernfun:NMvectors','N and M must be vectors.') .eG_>2'1
end R^tDL
~"i4"Op&
if length(n)~=length(m) ^y3snuLtE
error('zernfun:NMlength','N and M must be the same length.') /|aD,JVN"
end AJR`ohh
T`SpIdzB.
n = n(:); f>jAu;S
m = m(:); ip2BvN&
if any(mod(n-m,2)) Ah1fcXED
error('zernfun:NMmultiplesof2', ... 9xIz[`)i.
'All N and M must differ by multiples of 2 (including 0).') g;t>jgX
end -`} d@x
F}{uY(hv"[
if any(m>n) |(O _K(
error('zernfun:MlessthanN', ... 2^T`> ?{X
'Each M must be less than or equal to its corresponding N.') a^.5cJ$]
end 9{XC9\~
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if any( r>1 | r<0 ) ylBjuD+
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @#KZ2^
end GsvB5i
FvV:$V|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V~_aM@q1
error('zernfun:RTHvector','R and THETA must be vectors.') !%Qm{R
end ucgp=bye
"[p-Iy1
r = r(:); 18kzR6(W
theta = theta(:); ieG%D
HN
length_r = length(r); =r1@?x
if length_r~=length(theta) n0<I
error('zernfun:RTHlength', ... KiO1l{.s8n
'The number of R- and THETA-values must be equal.') t&L+]I'P3
end |XoW
Z,K
k\`~v$R3
% Check normalization: )TV{n#n
% -------------------- X!ad~bt
if nargin==5 && ischar(nflag) S6bW?8`
isnorm = strcmpi(nflag,'norm'); x cA5
if ~isnorm k^v P|*eu
error('zernfun:normalization','Unrecognized normalization flag.') Fi_JF;
end j1U,X
else *mTx0sQz(J
isnorm = false; =&xNdc
end y7WO:X&
N)b.$aC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MW$
X4<*KD
% Compute the Zernike Polynomials T`gR&n<D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BA t0YE`-,
v6q oH)n
% Determine the required powers of r: -OPJB:7Z
% -----------------------------------
*aT\V64
m_abs = abs(m); u?+i5=N9{
rpowers = []; YqSkz|o}m
for j = 1:length(n) Y}Gf%Xi,
rpowers = [rpowers m_abs(j):2:n(j)]; "g>, X[g
end -VVJf5/
rpowers = unique(rpowers); I#U"DwM
XlGDv*d:#d
% Pre-compute the values of r raised to the required powers, LIZsDTU
% and compile them in a matrix: `bx}!;{lx
% ----------------------------- z c7P 2@
if rpowers(1)==0 0.}WZAYy~
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]E!b&
rpowern = cat(2,rpowern{:}); 01/yog
rpowern = [ones(length_r,1) rpowern]; FyV)Nmc%t
else Mp`2[S@$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Bp>Z?"hTe
rpowern = cat(2,rpowern{:}); cgevP`*]
end u>W:SM
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c#_
% Compute the values of the polynomials: 3(n+5~{e
% -------------------------------------- aS,M=uqqK
y = zeros(length_r,length(n)); ;+-M+9"?O
for j = 1:length(n) mxQPOu
s = 0:(n(j)-m_abs(j))/2; *8?0vkZZ2
pows = n(j):-2:m_abs(j); m^M sp:T,
for k = length(s):-1:1 /$NZj"#
p = (1-2*mod(s(k),2))* ... ]= nM|e
prod(2:(n(j)-s(k)))/ ... u|}p3-z|Y
prod(2:s(k))/ ... B(TE?[ #
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K~DQUmU@
prod(2:((n(j)+m_abs(j))/2-s(k))); o,yP9~8\
idx = (pows(k)==rpowers); SZ'2/#R>
y(:,j) = y(:,j) + p*rpowern(:,idx); 9C_Vb39::$
end gJUawK
xYUC|c1Q9
if isnorm %SHgXd#X
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); mv;;0xH
end :G\X
end :t8?!9g
% END: Compute the Zernike Polynomials 1U(P0$C
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $63_*9
ta'{S=^j
% Compute the Zernike functions: -mur`tC
% ------------------------------ lUJ~_`D
idx_pos = m>0; ;Or]x?-
idx_neg = m<0; H;.${u^lhd
,6iXl ch
z = y; #5b}"xK{
if any(idx_pos) n#Y=y#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); N!#0O.6
end X}@'FxIF
if any(idx_neg) +8#hi5e
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E[8R
)xC@
end f+xhS,iDR
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% EOF zernfun