This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.

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This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue-Nagell equation x 2 + D = y n , x, y integers, n 3, for D in the range 1 D 100.

Abstract-This paper is concerned with hierarchical Markov random field (MRF) models and their application to sonar image segmentation. We present an original hierarchical segmentation procedure devoted to images given by a high-resolution sonar. The sonar image is segmented into two kinds of regions: shadow (corresponding to a lack of acoustic reverberation behind each object lying on the sea-bed) and sea-bottom reverberation. The proposed unsupervised scheme takes into account the variety of the laws in the distribution mixture of a sonar image, and it estimates both the parameters of noise distributions and the parameters of the Markovian prior. For the estimation step, we use an iterative technique which combines a maximum likelihood approach (for noise model parameters) with a least-squares method (for MRF-based prior). In order to model more precisely the local and global characteristics of image content at different scales, we introduce a hierarchical model involving a pyramidal label field. It combines coarse-to-fine causal interactions with a spatial neighborhood structure. This new method of segmentation, called scale causal multigrid (SCM) algorithm, has been successfully applied to real sonar images and seems to be well suited to the segmentation of very noisy images. The experiments reported in this paper demonstrate that the discussed method performs better than other hierarchical schemes for sonar image segmentation.

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