27-07-2012, 03:12 PM
Image Restoration via Nonstandard Diusion
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Abstract
We present a functional of nonstandard growth for which the corresponding minimization problem provides a
model for image denoising, enhancement, and restoration. The di®usion resulting from the proposed model is
a combination of isotropic and anisotropic di®usion. Isotropic di®usion is used at locations with low gradient
and total variation based di®usion is used along likely edges. At all other locations, the type of anisotropy varies
according to the local image information. Experimental results illustrate the e®ectiveness of the model in removing
noise and retaining sharp edges while avoiding the 'staircasing e®ect'. Existence and uniqueness of the proposed
model are also established.
Introduction
Over the past 20 years, the use of variational methods and nonlinear partial di®erential equations (PDEs) has
signi¯cantly grown and evolved to address the image restoration problem. Here we consider image restoration as
the classical inverse problem in which a piecewise smooth image is recovered from noisy data. The challenging aspect
of this problem is to design methods which can selectively ¯lter extraneous information, such as noise, without
losing signi¯cant features or creating false ones. Many nonlinear models have been proposed for this purpose,
however, when an image consists of objects of nonuniform intensity or has been degraded by noise, some of the
most successful noise removal techniques which retain and even enhance sharp edges often exhibit a 'staircasing
e®ect'. This can result in the generation of false edges which may in turn yield an incorrect segmentation. Our
goal is to study a new model for image restoration which not only removes noise and retains sharp edges, but also
avoids staircasing in what should be smooth regions.
Current PDE-based image restoration models can be classi¯ed into three types: curvature driven di®usion,
tensor driven di®usion, and variational methods. Curvature based models di®use only along the level curves of an
image u (in particular, not at all in the direction ru), so edges are preserved in the denoising process. The speed
of the di®usion depends on the local curvature, and is often penalized where the magnitude of the image gradient
is large [2, 8, 13, 15, 16, 18, 19, 20, 25]. Tensor driven di®usion is governed by a matrix built into the di®usion
equation [10, 12, 28, 29]. The di®usion governed by this type of model is anisotropic and the matrix determines
the direction(s) of the di®usion as well as the speed in each direction. The main feature of this model is that the
eigenvalues of the matrix can be chosen so the model can enhance speci¯c features such as edges or textures.
A large number of image restoration techniques are conveniently formulated using a variational approach. The
Mumford-Shah functional [17] is often used as a prototype for the free discontinuity problem. When used for image
restoration and segmentation, the main characteristic of the Mumford-Shah model is that it di®uses isotropically
while minimizing the lengths of the edges. The classic Rudin-Osher-Fatemi model [22] which minimizes the total
variation of an image yields di®usion strictly along the level curves of an image. Models based on this method
are very successful at recovering piecewise constant images with sharp edges since di®usion is only in the direction
orthogonal to ru [21, 22, 24, 25, 26, 27].
For images where objects are represented by non-uniform intensities, edges cannot be de¯ned as the boundaries
of homogeneous regions. Furthermore, in highly degraded images, di®usion which is strictly in one direction may
create false edges, the phenomenon often referred to as 'staircasing' (see ¯gures 2 and 3). In these cases, one may
want more °exibility in both the direction and speed of the di®usion. One solution is to di®use isotropically away
from edges and anisotropically near likely edges. This feature does not occur in any of the three above mentioned
classes of models and although it has been explored in the literature, [9, 3, 23, 4, 5, 7], often either the interpolation
between isotropic and anisotropic di®usion is di±cult to control, or the mathematical foundations are di±cult to
establish.
In this paper we propose a new image reconstruction model which preserves ¯ne structures and object boundaries
with low gradient while avoiding the 'staircasing e®ect' in piecewise smooth images. This is done using anisotropic
di®usion which is between isotropic and total variation (TV) based. The type of anisotropy depends on the local
image information, thus providing a natural control of the interpolation. We use a variational approach, as it
allows us to express the model using a concise formulation which can be studied mathematically and implemented
using straightforward ¯nite di®erence methods.
The paper is organized as follows. In section 2 we discuss models which combine isotropic and TV-based di®usion
and introduce the proposed model. Our numerical schemes are presented in section 3 and experimental results in
section 4. Section 5 is our concluding remarks.