a la physique de l'information - Lisa - Université d'Angers

Constructive action of noise for impulsive
noise removal in scalar images
A. Histace and D. Rousseau
A nonlinear variational approach to remove impulsive noise in scalar
images is proposed. Taking benefit from recent studies on the use of
stochastic resonance and the constructive role of noise in nonlinear
processes, the process is based on the classical restoration process of
Perona-Malik in which a Gaussian noise is purposely injected. It is
shown that this new process can outperform the original restoration
process of Perona-Malik.
Aim and motivation: Removing impulsive noises from scalar images
is a problem of great interest since these short duration and high
energy noises can degrade the quality of digital images in a large
variety of practical situations [1]. In this context of non-Gaussian
noise, nonlinear processes are often invoked. Among these nonlinear
processes median filtering is a classical tool leading to good results
[2]. Nevertheless, these median filtering techniques involve strong
statistics calculation and can turn out to be highly time consuming to
compute. Another nonlinear process classically used for restoration
tasks is the diffusion process of Perona-Malik [3]. This process, based
on a variational approach, presents short implementation time and has
the ability to remove noise while keeping edges stable on many scales.
The Perona-Malik process also has its own limitations. Among these
limitations, the smoothing property of the diffusion process does not
preserve the information present in areas with texture or small but
significative gradients [4]. As paradoxical as it may seem, to limit the
effect of this drawback, we propose a new variant of the Perona-Malik
process in which a controlled amount of noise is injected into the
nonlinear process. The possibility of constructive action of noise in
nonlinear processes is now a well established paradigm known under
the name of stochastic resonance (see [5] for a recent overview).
To date, this paradigm has essentially been illustrated with monodimensional
signals. This work is a new feature of noise enhanced
information processing presented here for the first time in the context
of image restoration.
Method: Let cori denote an original image and c0 denote the same
image corrupted by an input impulsive noise x imposed by the
external environment:
c0ðx; yÞ ¼coriðx; yÞþxðx; yÞ ð1Þ
The restoration of c0 aims at the removal of x from c0 to obtain an
image as similar as possible to cori of (1).
The Perona-Malik restoration approach of c0 is equivalent to an
iterative minimisation problem [6], solved by the resolution of the
partial differential equation (PDE) given by:
@cðx; y; tÞ
¼ divðgðjHcðx; y; tÞjÞHcðx; y; tÞÞ; cðx; y; t ¼ 0Þ ¼c0 @t
ð2Þ
where g(.) is a nonlinear decreasing function of the gradient (H)ofc the
restored image at a scale t (which can be interpreted as a time evolution
parameter) and div is the divergence operator. For practical numerical
implementation, the process of (2) is discretised with a time step t. The
images c(t n) are calculated, with (2), at discrete instant t n ¼ nt with n the
number of iterations in the process. We now compare the standard
Perona-Malik process of (2) with the following diffusion process
@cðx; y; tÞ
¼ divðg
@t
ZðjHcðx; y; tÞjÞHcðx; y; tÞÞ ð3Þ
in which the nonlinear function g(.) in (2) has been replaced by g Z(.) with
gZðuðx; yÞÞ ¼ 1
M
P M
i¼1
gðuðx; yÞþZ i ðx; yÞÞ ð4Þ
where Z i functions are M independent noises assumed independent
and identically distributed with probability density function (PDF) f Z
and RMS amplitude s Z. The noises Z i which are purposely added noises
applied to influence the operation of the g(.) have to be clearly
distinguished from the input noise x of (1), which is considered as a
noise imposed by the external environment that we wish to remove. The
ELECTRONICS LETTERS 30th March 2006 Vol. 42 No. 7
choice of g Z in (4) is inspired by recent studies on the constructive
action of noise in parallel arrays of nonlinear electronic devices [7] and
transposed here in the domain of image processing. The quality of the
restored image c(tn) at a given instant tn is assessed by the normalised
cross-covariance C coric(tn ) given by:
CcoricðtnÞ ¼ hðcori hcoriiÞðcðtnÞ hcðt
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nÞiÞi
hðcori hcoriiÞ 2 ihðcðtnÞ hcðtnÞiÞ 2 q ð5Þ
i
where h..i denotes the spatial average.
Results: For illustration of the processes of (2) and (3), the image
‘cameraman’ (see image a in Fig. 2), which presents strong and small
gradients, textured and non-textured regions of interest, has been
taken as the reference image in this study.
Fig. 1 Normalised cross-covariance of (5) against iteration number n of
restoration process
– – – our modified version of Perona-Malik process of (3) and (4) for various
number M of independent noises Zi ——— classical Perona-Malik process of (2)
RMS amplitude sZ of M noises Zi fixed to sZ ¼ 0.7. Parameter k in nonlinear
function g(.) and step time t, characterising convergence speed of the diffusion
process, fixed to k ¼ 0.2 and t ¼ 0.25 in both (2) and (3)
a
b
Fig. 2 Visual comparison of performance of restoration processes by (2)
and (3)
a: original ‘cameraman’ image cori of (1). b: noisy version c0 of cori corrupted
by additive salt and pepper noise x (1) of standard deviation 0.1. c and d obtained
with (2) and (3), respectively, for optimal number of iterations n corresponding to
highest value of normalised cross-covariance in Fig. 1 (M ¼ 11 in the case of d)
The original nonlinear function g(.) proposed by Perona-Malik in [3],
with g(u(x, y)) ¼ e u(x,y)=k2
, is chosen in this study. The PDF f Z of the M
c
d
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