Analytic torsion of complete hyperbolic manifolds of finite volume

In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.Comment: 56 page

Selberg and Ruelle zeta functions and the relative analytic torsion on complete odd-dimensional hyperbolic manifolds of finite volume

Let X be a complete hyperbolic manifold of finite volume and of odd dimension d. Firstly, we study Selberg zeta functions on X, prove that these functions have a meromorphic continuation to the entire complex plane and describe their singularities. Secondly, we define the relative or regularized analytic torsion of X associated to certain representations of its fundamental group. We investigate the asymptotic behaviour of this torsion with respect to special sequences of representations. Finally, if X is 3-dimensional, we establish a relation between the regularized analytic torsion and the behaviour of a twisted Ruelle zeta function at 0. Our work generalizes results of Fried, Bunke and Olbrich, Bröcker and Wotzke to the non-compact case and results of Müller to the non-compact and higher-dimensional situation

Region-Based Template Matching Prediction for Intra Coding

Copy prediction is a renowned category of prediction techniques in video coding where the current block is predicted by copying the samples from a similar block that is present somewhere in the already decoded stream of samples. Motion-compensated prediction, intra block copy, template matching prediction etc. are examples. While the displacement information of the similar block is transmitted to the decoder in the bit-stream in the first two approaches, it is derived at the decoder in the last one by repeating the same search algorithm which was carried out at the encoder. Region-based template matching is a recently developed prediction algorithm that is an advanced form of standard template matching. In this method, the reference area is partitioned into multiple regions and the region to be searched for the similar block(s) is conveyed to the decoder in the bit-stream. Further, its final prediction signal is a linear combination of already decoded similar blocks from the given region. It was demonstrated in previous publications that region-based template matching is capable of achieving coding efficiency improvements for intra as well as inter-picture coding with considerably less decoder complexity than conventional template matching. In this paper, a theoretical justification for region-based template matching prediction subject to experimental data is presented. Additionally, the test results of the aforementioned method on the latest H.266/Versatile Video Coding (VVC) test model (version VTM-14.0) yield an average Bjøntegaard-Delta (BD) bit-rate savings of −0.75% using all intra (AI) configuration with 130% encoder run-time and 104% decoder run-time for a particular parameter selection