Biomedical Engineering Reference
In-Depth Information
Chapter 4
Sparse Bayesian (Champagne) Algorithm
4.1 Introduction
In this chapter, we provide a detailed description of an algorithm for electromagnetic
brain imaging, called the Champagne algorithm [
1
,
2
]. The Champagne algorithm
is formulated based on an empirical Bayesian schema, and can provide a sparse
solution, since the sparsity constraint is embedded in the algorithm. The algorithm is
free from the problems that cannot be avoided in other sparse-solution methods, such
as the
L
1
-regularized minimum-norm method. Such problems include the difficulty
in reconstructing voxel time courses or the difficulty in incorporating the source-
orientation estimation.
using the Gaussian prior for the
j
th voxel value,
1
, ʱ
−
1
x
j
∼
N(
x
j
|
0
),
(4.1)
where the precision
is common to all
x
j
. In this chapter, we use the Gaussian prior
whose precision (variance) is specific to each
x
j
, i.e.,
ʱ
, ʱ
−
1
j
x
j
∼
N(
x
j
|
0
).
(4.2)
We show that this “slightly different” prior distribution gives a solution totally dif-
ferent from the
L
2
-norm solution. Actually, the prior distribution in Eq. (
4.2
) leads to
a sparse solution. The estimation method based on the prior in Eq. (
4.2
) is called the
sparse Bayesian learning in the field of machine learning [
3
,
4
], and the source recon-
struction algorithm derived using Eq. (
4.2
) is called the Champagne algorithm [
1
].
In this chapter, we formulate the source reconstruction problem as the spatiotem-
poral reconstruction, i.e., the voxel time series
x
1
,
x
2
,...,
x
K
is reconstructed using
the sensor time series
y
1
,
are denoted
y
k
and
x
k
. We use collective expressions
x
and
y
, indicating the whole voxel time series
y
2
,...,
y
K
where
y
(
t
k
)
and
x
(
t
k
)
1
We use the notational convenience
N
(
variable
|
mean
,
covariance matrix
)
throughout this topic.
