MRI TV-Rician Denoising - Biomedical Engineering Systems and Technologies - page 232

are acquired). When the resulting functional is considered for minimization, the varia-
tional approach leads to the resolution of a nonlinear degenerate PDE elliptic equation
as the Euler Lagrange equation for optimization. This has a number of theoretical prob-
lems when the Total Variation operator is considered as a smoother, because the energy
functional is not differentiable at the origin (i.e. u = 0 ) and regular, approximating

problems must be solved. In turn this approach cause a over-smoothing effect in the
numerical solutions of the model and accuracy in fine scale details is lost because the
edges diffuse. A direct gradient descent method has been used in [4] in order to validate
the model assumption of Rician noise but the method is found to be inherently slow
because a stabilization at the steady state is needed. Also, that scheme is finally explicit
and very small time steps have to be used to avoid numerical oscillations.
Our aim is to present a new framework to solve numerically and efficiently the gradi-
ent descent scheme (gradient flow) associated to the Rician energy minimization prob-
lem introducing a new semi-implicit formulation. Using a simple Euler discretization
of the time derivative, stationary problems of the Rudin, Osher and Fatemi (ROF) type
[7] are deduced. This allows to use the well known dual formulation of the ROF model
proposed in [8] for a speed up of the computations. As a by-product of this approach
the exact Total Variation operator can be computed and this provides accuracy to the
solution in so far truly (discontinuous) bounded variation solutions are numerically ap-
proximated.
This paper is organized as follows: in section 2 and 3 we present the model equa-
tion and the numerical scheme recently proposed in [4]. In section 4 we propose a new
framework which leads to a more efficient and accurate numerical scheme. The pro-
posed method is tested in section 5, where we consider synthetic MR brain images to
compare it with the method of [4] and some preliminary results of applying this algo-
rithm to real Diffusion Weighted Magnetic Resonance Images (DW-MRI) are shown in
subsection 5. Finally in section 6 we present our conclusions.

∇

2 Model Equations

d , d =2 , 3 defining the image domain and let

Let Ω be a bounded open subset of R

L ∞ ( Ω )

f : Ω [0 , 1]

(otherwise we normalize). Let BV ( Ω ) be the space of functions with bounded variation
in Ω equipped with the seminorm

→ R be a given noisy image representing the data, with f ∈ ∩

| u | BV defined as

| BV = sup Ω u ( x ) div ξ ( x ) dx :

ξ ξ ( x )

C c ( Ω,

d ) ,

| u ∈ R | | ∞ ≤ 1 ,x ∈ (1)

Ω

ξ

d ,

where (details on this space and the

related geometric measure theory can be found in [9]). Following a Bayesian modelling
approach we consider the minimization problem

|·| ∞ denotes the l ∞ norm in R | | ∞ =max

1 ≤i≤d |

ξ i |

min J ( u )+ λH ( u,f ) (2)

u ∈ BV ( Ω )

where J ( u ) is the convex nonnegative total variation regularization functional