Spatially Adaptive Intensity Bounds for Image Restoration
EURASIP Journal on Applied Signal Processing
Spatially Adaptive Intensity Bounds for Image Restoration
Kaaren L. May 0 1 2
Aggelos K. Katsaggelos
0 Department of Electrical and Computer Engineering, Northwestern University , Evanston, IL 60208-3118 , USA
1 Communications and Signal Processing Group, Department of Electrical and Electronic Engineering, Imperial College London , London SW7 2BT , UK
2 Snell and Wilcox Ltd., Liss Research Centre , Liss Mill, Mill Road, Liss, Hampshire, GU33 7BD , UK
Spatially adaptive intensity bounds on the image estimate are shown to be an effective means of regularising the ill-posed image restoration problem. For blind restoration, the local intensity constraints also help to further define the solution, thereby reducing the number of multiple solutions and local minima. The bounds are defined in terms of the local statistics of the image estimate and a control parameter which determines the scale of the bounds. Guidelines for choosing this parameter are developed in the context of classical (nonblind) image restoration. The intensity bounds are applied by means of the gradient projection method, and conditions for convergence are derived when the bounds are refined using the current image estimate. Based on this method, a new alternating constrained minimisation approach is proposed for blind image restoration. On the basis of the experimental results provided, it is found that local intensity bounds offer a simple, flexible method of constraining both the nonblind and blind restoration problems.
and phrases; image resolution; blur identification; blind image restoration; set-theoretic estimation
1. INTRODUCTION
In many imaging systems, blurring occurs due to factors such
as relative motion between the object and camera,
defocusing of the lens, and atmospheric turbulence. An image may
also contain random noise which originated in the formation
process, the transmission medium, and/or the recording
process.
The above degradations are adequately modelled by a
linear space-invariant blur and additive white Gaussian noise,
yielding the following model:
g = h ∗ f + v,
(1)
where the vectors g, f , h, and v correspond to the
lexicographically ordered degraded and original images, blur, and
additive noise, respectively, which are defined over an
array of pixels (m, n). The two-dimensional convolution can
be expressed as h ∗ f = Hf = Fh, where H and F are
block-Toeplitz matrices and can be approximated by
blockcirculant matrices for large images [1, Chapter 1].
The goal of image restoration is to recover the
original image f from the degraded image g. In classical image
restoration, the blur is known explicitly prior to restoration.
However, in many imaging applications, it is either costly
or physically impossible to completely characterise the blur
based on a priori knowledge of the system [
2
]. The recovery
of an image when the blur is partially or completely unknown
is referred to as blind image restoration. In practice, some
information about the blur is needed to restore the image.
There are a number of factors which contribute to the
difficulty of image restoration. The problem is ill posed in
the sense that if the image formation process is modelled in
a continuous, infinite-dimensional space, then a small
perturbation in the output, that is, noise, can result in an
unbounded perturbation of the least squares solution of (1) for
the image or the blur [
1
]. Although the discretised inverse
problem is well posed [
3
], the ill-posedness of the continuous
problem leads to the ill-conditioning of H or F. Therefore,
direct inversion of either matrix leads to excessive noise
amplification, and regularisation is needed to limit the noise in
the solution.
The blind image restoration problem is also ill defined,
since the available information may not yield a unique
solution to the corresponding optimisation problem. Even if a
unique solution exists, the cost function is, with the
exception of the NAS-RIF algorithm [
4
], nonconvex, and
convergence to local minima often occurs without proper
initialisation. Undesirable solutions can be eliminated by
incorporating more effective constraints.
In this paper, spatially adaptive intensity bounds are
therefore proposed as a means of (1) regularising the
illposed restoration problem; and (2) limiting the solution
space in blind restoration so as to avoid convergence to
undesirable solutions. The bounds are implemented in the
framework of the gradient projection method proposed in [
5, 6
].
Prior research on spatially adaptive intensity bounds
has been conducted solely in the context of classical image
restoration. Local intensity bounds were first introduced in
[
7
] for artifact suppression. These bounds were applied to
the Wiener filtered image, that is, they were applied to a
solution rather than to the optimisation problem itself. In [
8
], it
was shown that the constraints could be incorporated in the
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