3.1.1
Consider the problem of restoring a signal f from the data d=f+e
where e denotes the noise. The regularization formulation defines the
solution 
to be the global minimum of an energy function 
,
. The energy is the sum of two terms
The closeness term, 
, measures the cost
caused by the discrepancy between the solution f and the data d
where 
is a weighting function and a and b are the bounds
of the integral. The smoothness term, 
, measures the cost
caused by the irregularities of the solution f, the irregularities
being measured by the derivative magnitudes 
. With
identical independent additive Gaussian noise, 
, 
and 
correspond to the energies in the posterior, the
likelihood the prior Gibbs distributions of an MRF, respectively
[Marroquin et al. 1987].
The smoothness term 
, also called a regularizer, is the
object of study in this work. It penalizes the irregularities according
to the a priori smoothness constraint encoded in it. It is
generally defined as
where 
is the 
order regularizer, N is the highest
order to be considered and 
is a weighting factor. A
potential function 
is
the penalty against the irregularity in 
and corresponds
to prior clique potentials in MRF models. Regularizers differ in the
definition of 
, more specifically in the selection of g.
