Th`ese Marouan BOUALI - Sites personnels de TELECOM ParisTech

73
Chapitre 4
A Variational approach for the
destriping issue
4.1 PDEs and variational methods in image processing
Many disciplines from physical sciences such as thermodynamics and fluids mechanics
have inspired the application of Partial Differential Equations (PDEs) in the field
of image processing. Early denoising methods are mostly based on smoothing operations,
either computed directly in the spatial domain via a convolution with a filter, or in the
frequency domain. Assuming the noise to be contained in the high frequencies of the observed
signal, a common restoration technique consists in convolving the noisy image with
a linear operator. Denoting u 0 the noisy signal defined in a bounded domain Ω of R 2 , an
estimate of the true image u is obtained by considering the scale-space generated by u 0
as :
∫
u(x, y, t) = G(x − ξ, y − η, t)u 0 (ξ, η)dxdy, (x, y) ∈ Ω (4.1)
Ω
Typically, the operator G is a bidimensional gaussian kernel :
G(x, y, t) = 1 ( −(x 2
4πt exp + y 2 )
)
4t
(4.2)
where the variance σ 2 =2t of the gaussian operator controls the degree of smoothing in
the restored image u. The work of [Koenderink, 1984] reformulates the convolution with
a gaussian kernel as a diffusion process where the value of a pixel can be expressed with
respect to a neighboorhood which size is determined by the variance σ 2 of the operator
G. The noise free image u can be seen as the solution of a simple parabolic PDE :
∂u
∂t (x, y, t) u
=∂2 ∂ 2 x (x, y, u
t)+∂2 ∂ 2 (x, y, t)
y
u(x, y, 0) = u 0 (x, y)
(4.3)