MRI TV-Rician Denoising - Biomedical Engineering Systems and Technologies - page 234

their Gibb's prior model based on the Perona and Malik energy functional [11] . Also it
differs from the classical gaussian noise model because of the nonlinear dependence of
the solution of the ratio I 1 /I 0 .

3 The Primal Descent Gradient Numerical Scheme

A number of mathematical difficulties is associated with the multivalued formulation
(7) and a regularization of the diffusion term div ( ∇ u/ |∇ u | ) in form div ( ∇ u/ |∇ u | ) ,

| = |∇

with |∇ u u | 2 + 2 and 0 < 1 is implemented to avoid degeneration of

u = 0 . Using this approximation it is possible to give a (weak)
meaning to the following formulation:
Fixed λ , σ and (small) and given f

the equation where ∇

L ∞ ( Ω ) W 1 , 1 ( Ω )

∈ ∩ [0 , 1] find u ∈ ∩ [0 , 1]

solving

div ∇

+ λ

u

−

σ 2 [ u −

r ( u ,f ) f ]=0

(8)

|∇ u |

complemented with Neumann homogeneous boundary conditions ∂u /∂n =0 and
where, for notational simplicity, we introduced the nonlinear function r ( u ,f )=
I 1 ( u f/σ 2 ) /I 0 ( u f/σ 2 ) .
This is a nonlinear (in fact quasilinear) elliptic problem that we solve with a gradient
descent scheme until stabilization (when t

) of the evolutionary solution to
steady state, i.e. a solution of the elliptic problem (8) which is a minimum of the energy

→ + ∞

J ( u )+ λH ( u ,f )=

uf
σ 2

dx

=
Ω

u 2 dx − λ
Ω

λ
2 σ 2

|∇u | 2 + 2 dx + log I 0 (9)

Ω

When → 0 we have u → u , J ( u ) → J ( u ) and the energies in (6) and (9) coincide.
This approach amounts to solve the associated nonlinear parabolic problem:

= div ∇

∂u
∂t

u λ

σ 2

− [ u − r ( u ,f ) f ] (10)

|∇ u |

complemented with Neumann homogeneous boundary conditions ∂u /∂n =0 and
initial condition u (0 ,x )= u

)
to the steady state of (8), i.e. a minimum of (9) which approximates, for sufficiently
small, a minimum of the energy functional (6). Following [4] and using forward fi-
nite difference for the temporal derivative it is straightforward to define a semi-implicit
iterative scheme which simplifies to the explicit one:
1+ τ λ
σ 2

( x ) whose (weak) solution stabilizes (when t → + ∞

0

u n +1

= u n + τ div ∇

+ λ
σ 2

r ( u n ,f ) f

u n

(11)

|∇ u n |

where τ is the time step and spatial discretization for the approximated TV-term is
performed as in [12] .