Minimum-Norm-Based Source Imaging Algorithms - Electromagnetic Brain Imaging - page 26

ʣ y is expressed as

where according to Eq. (B.30),

ʣ y = ʲ − 1 I + ʱ − 1 FF T . (2.65)

If the increase of the likelihood in Eq. ( 2.64 ) with respect to the iteration count
becomes very small, the iteration may be stopped.

2.10.3 L 1 -Regularized Method

The method of L 1 -norm regularization can also be derived based on the Bayesian
formulation. To derive the L 1 -regularization. we use the Laplace distribution as the
prior distribution

2 b exp

N

1 1

b |

p ( x ) = − x j | . (2.66)

j = 1

Then, using Eq. ( 2.57 ), (and replacing F with H ), the cost function is derived as

N

2

F( x ) = ʲ y − Hx + 2 b

1 |

x j | , (2.67)

j =

ʾ = /ʲ

which is exactly equal to Eq. ( 2.53 ), if we set .

Another formulation for deriving the L 1 -regularized method is known. It uses the
framework of the sparse Bayesian learning described in Chap. 4 . In Chap. 4 , assuming
the Gaussian prior,

2 b

ʱ j
2

1 / 2 exp

2 x j

N N

− ʱ j , ʱ − 1

j

p ( x | ʱ ) =

1 N(

x j | 0 ) = , (2.68)

ˀ

j = j = 1

we derive the marginal likelihood for the hyperparameter ʱ =[ ʱ 1 ,...,ʱ N ] ,

p ( y | ʱ ) , using,

p ( y | ʱ ) = p ( y | x ) p ( x | ʱ ) d x , (2.69)

and eventually derive the Champagne algorithm. However, instead of implementing
Eq. ( 2.69 ), there is another option in which we compute the posterior distribution
p

( | )

x y using

p ( x | y ) ∝ p ( y | x ) p ( x | ʱ ) p ( ʱ ) d ʱ = p ( y | x ) p ( x ), (2.70)