Spatially Adaptive Intensity Bounds for Image Restoration
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Spatially Adaptive Intensity Bounds for Image Restoration Kaaren L. May Snell and Wilcox Ltd., Liss Research Centre, Liss Mill, Mill Road, Liss, Hampshire, GU33 7BD, UK Email: [email protected]
Tania Stathaki Communications and Signal Processing Group, Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2BT, UK Email: [email protected]
Aggelos K. Katsaggelos Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL 60208-3118, USA Email: [email protected] Received 9 December 2002 and in revised form 24 June 2003 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. Keywords 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 re
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