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We study the tensor product decomposition of the split real quantum group Uqq~(sl(2,R)) from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of Uqq~(sl(2,R)) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group Uq(sl2) by solving certain functional equations and using normalization arising from tensor products of canonical basis. We propose a general strategy to deal with the tensor product decomposition for the higher rank split real quantum group Uqq~(gR)

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and q-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

The joint bilateral filter, which enables feature-preserving signal smoothing according to the structural information from a guidance, has been applied for various tasks in geometry processing. Existing methods either rely on a static guidance that may be inconsistent with the input and lead to unsatisfactory results, or a dynamic guidance that is automatically updated but sensitive to noises and outliers. Inspired by recent advances in image filtering, we propose a new geometry filtering technique called static/dynamic filter, which utilizes both static and dynamic guidances to achieve state-of-the-art results. The proposed filter is based on a nonlinear optimization that enforces smoothness of the signal while preserving variations that correspond to features of certain scales. We develop an efficient iterative solver for the problem, which unifies existing filters that are based on static or dynamic guidances. The filter can be applied to mesh face normals followed by vertex position update, to achieve scale-aware and feature-preserving filtering of mesh geometry. It also works well for other types of signals defined on mesh surfaces, such as texture colors. Extensive experimental results demonstrate the effectiveness of the proposed filter for various geometry processing applications such as mesh denoising, geometry feature enhancement, and texture color filtering.

Detecting deformities on objects is a typical topic in shape analysis, and has much applications such as abnormalities detection in medical imaging (e.g. growth of tumor, spread of cancer). While many algorithms are already well-established in a 2-dimensional case when the object is indeed a surface, a model that still performs well in the general n-dimensional case is still missing. It is our goal in this paper to complete this missing piece, by introducing an indicator in order to effectively distinguish between normal and abnormal deformities. The proposed framework is closely related to the classic 2-dimensional conformal geometry and quasi-conformal geometry. In this work, we model abnormal deformations by anisotropic deformations. Given any two objects of the same dimension (with landmark constraints in between), we define the Anisotropic Indicator, a locally defined real-valued function on the original object, which demonstrates the abnormalities in the deformation between them. Both global and local features about the abnormalities between the two objects can be tracked by analyzing the indicator. We tested the algorithm by detecting deformations on synthetic data and real data, and results show that our algorithm can detect deformations of different types and degrees.