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Orthogonal Matching Pursuit---OMP算法描述
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Orthogonal Matching Pursuit-Recursive Function Approximation with Applications to wavelet decomposition, OMP算法描述
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Orthogonal Matching Pursuit:
Recursive Function Approximat ion with Applications to Wavelet
Decomposition
Y.
C.
PATI
Information Systems Laboratory
Dept.
of
Electrical Engineering
Stanford University, Stanford,
CA
94305
R.
REZAIIFAR
AND
P.
s.
KRISHNAPRASAD
Institute for Systems Research
Dept.
of
Electrical Engineering
University
of
Maryland, College
Park,
MD
20742
Abstract
where
fk
is the current approximation, and
Rk
f
the
current residual (error). Using initial values of
Ro
f
=
f
,
fo
=
0,
and
k
=
1
,
the
MP
algorithm is comprised
of the following steps,
In this paper we describe
a
recursive algorithm to
compute representations of functions with respect to
nonorthogonal and possibly overcomplete
dictionaries
(I) Compute the inner-products
{(Rkf,
Zn)},.
of elementary building blocks
e.g.
affine (wavelet)
frames. We propose
a
modification to the Matching
Pursuit algorithm of Mallat and Zhang
(1992)
that
maintains full backward orthogonality of the residual
convergence. We refer to this modified algorithm
as
Orthogonal Matching Pursuit (OMP). It is shown that
all additional computation required for the OMP al-
gorithm may be performed recursively.
(11) Find
nktl
such that
(error) at every step and thereby leads to improved
I(Rkf,
'nk+I)l
2
asYp
I(Rkf,
zj)I
1
where
0
<
a
5
1.
(111) Set,
1
Introduction and Background
Given
a
collection of vectors
D
=
{xi}
in a Hilbert
(IV) Increment
k, (k
t
IC
+
l),
and repeat steps
(1)-
(IV) until some convergence criterion has been
satisfied.
space
R,
let us define
V=Span{z,}, and
W=V'
(inR).
We shall refer to
D
as
a
dictionary,
and will assume
the vectors
xn,
are normalized
(llznII
=
1).
In
[3]
Mal-
lat and Zhang proposed an iterative algorithm that
they termed Matching Pursuit (MP) to construct rep-
resentations of the form
Pvf
=
Canzn,
(1)
n
where
PV
is the orthogonal projection operator onto
V.
Each iteration of the MP algorithm results in an
intermediate representation of the form
1058-6393193
$03.00
Q
1993
IEEE
The proof of convergence
[3]
of MP relies essentially on
the
fact
that
(Rk+lf,
z~~+~)
=
0.
This orthogonality
of the residual to the last vector selected leads to the
following "energy conservation" equation.
IIRkfl12
=
llRk+lf112
+
I(Rkf12nk+1)12'
(2)
It has been noted that the MP algorithm may be de-
rived
as
a
special case of a technique known
as
Pro-
jection Pursuit
(c.f,
[2])
in the statistics literature.
A
shortcoming of the Matching Pursuit algorithm
in its originally proposed form is that although asymp
totic convergence is guaranteed, the resulting approxi-
mation after any finite number of iterations will in gen-
eral be suboptimal in the following sense. Let
N
<
00,
40
vct26
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