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VectorValued Image Regularization with PDE’s :
A Common Framework for Different Applications
D. Tschumperl´e
R. Deriche
INRIA, Odyss´ee Lab, 2004 Rte des Lucioles, BP 93, 06902 SophiaAntipolis, France.
{
David.Tschumperle,Rachid.Deriche
}
@sophia.inria.fr
Abstract
1
We address the problem of vectorvalued image regu
larization with variational methods and PDE’s. From the
study of existing formalisms, we propose a unifying frame
work based on a very local interpretation of the regulariza
tion processes. The resulting equations are then specialized
into new regularization PDE’s and corresponding numeri
cal schemes that respect the local geometry of vectorvalued
images. They are finally applied on a wide variety of image
processing problems, including color image restoration, in
painting, magnification and flow visualization.
1.
Anisotropic
regularization PDE’s
raise a strong interest in
the field of image processing. The ability to smooth data
while preserving large global features such as contours and
corners (discontinuities), has opened new ways to handle
classical imagerelated issues (restoration, segmentation).
Thus, many regularization schemes have been presented so
far in the literature, particularly for the case of
2
D
scalar
images
I
: Ω
⊂
R
2
→
R
( [1, 17, 18, 28] and references
therein).
Extensions of these algorithms to
vectorvalued
images
I
: Ω
→
R
n
have been recently proposed, leading
to more elaborated diffusion PDE’s : a
coupling between
image channels
appears in the equations, through the con
sideration of a
local vector geometry
, given pointwise by the
spectral elements
λ
+
, λ

(positive eigenvalues) and
θ
+
, θ

(orthogonal eigenvectors) of the
2
×
2
symmetric and semi
positivedefinite matrix
G
=
∑
n
j
=1
∇
I
j
∇
I
T
j
(also called
structure tensor
[25, 26, 28, 29]). The
λ
±
respectively de
fine the local min/max vectorvalued variations of
I
in cor
responding spatial directions
θ
±
, i.e.
the local configura
tion of the image discontinuities. (note that
λ
+
=
k∇
I
k
and
θ
+
=
∇
I/
k∇
I
k
for scalar images
,
n
= 1
). Regulariza
tion schemes generally lie on one of these three following
approaches, related to different interpretation levels :
(1) Functional minimization
: Regularizing an image
I
may be seen as the minimization of a functional
E
(
I
)
mea
suring a
global
image variation. The idea is that minimizing
this variation will flatten the image, then remove the noise :
min
I
:Ω
→
R
n
E
(
I
) =
R
Ω
φ
(
N
(
I
))
d
Ω
(1)
1
This article was published in CVPR2003  IEEE Conference on Com
puter Vision and Pattern Recognition , Madison/USA June 2003
where
φ
:
R
→
R
is an increasing function and
N
(
I
)
is a norm related to
local image variations
, for instance
N
(
I
) =
p
λ
+
+
λ

= trace (
G
)
1
2
.
The minimization of
(1) is performed with a gradient descent (PDE) given by the
EulerLagrange equations of
E
(
I
)
.
Useful references for
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