307 research outputs found
On the ergodicity of geodesic flows on surfaces of nonpositive curvature
Let be a smooth compact surface of nonpositive curvature, with genus
. We prove the ergodicity of the geodesic flow on the unit tangent
bundle of with respect to the Liouville measure under the condition that
the set of points with negative curvature on has finitely many connected
components. Under the same condition, we prove that a non closed "flat"
geodesic doesn't exist, and moreover, there are at most finitely many flat
strips, and at most finitely many isolated closed "flat" geodesics.Comment: 11 pages, 4 figures. Lemma 3.8 is added to correct a gap in the proof
of Proposition 3.4. Theorem 1.5 is also adde
Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
Let be a -partially hyperbolic diffeomorphism. We
introduce a type of modified Schmidt games which is induced by and played
on any unstable manifold. Utilizing it we generalize some results of \cite{Wu}
as follows. Consider a set of points with non-dense forward orbit: for some and for any . We show that
is a winning set for such modified Schmidt games played on ,
which implies that has Hausdorff dimension equal to .
Then for any nonempty open set we show that has
full Hausdorff dimension equal to , by using a technique of
constructing measures supported on with lower pointwise dimension
approximating .Comment: 19 pages. Remark 4.10 is corrected. We have followed the proof scheme
in \cite{Wu}. arXiv admin note: text overlap with arXiv:1311.530
Image Deblurring and Super-resolution by Adaptive Sparse Domain Selection and Adaptive Regularization
As a powerful statistical image modeling technique, sparse representation has
been successfully used in various image restoration applications. The success
of sparse representation owes to the development of l1-norm optimization
techniques, and the fact that natural images are intrinsically sparse in some
domain. The image restoration quality largely depends on whether the employed
sparse domain can represent well the underlying image. Considering that the
contents can vary significantly across different images or different patches in
a single image, we propose to learn various sets of bases from a pre-collected
dataset of example image patches, and then for a given patch to be processed,
one set of bases are adaptively selected to characterize the local sparse
domain. We further introduce two adaptive regularization terms into the sparse
representation framework. First, a set of autoregressive (AR) models are
learned from the dataset of example image patches. The best fitted AR models to
a given patch are adaptively selected to regularize the image local structures.
Second, the image non-local self-similarity is introduced as another
regularization term. In addition, the sparsity regularization parameter is
adaptively estimated for better image restoration performance. Extensive
experiments on image deblurring and super-resolution validate that by using
adaptive sparse domain selection and adaptive regularization, the proposed
method achieves much better results than many state-of-the-art algorithms in
terms of both PSNR and visual perception.Comment: 35 pages. This paper is under review in IEEE TI
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