Characterization of almost LpL^p-eigenfunctions of the Laplace-Beltrami operator

In \cite{Roe} Roe proved that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R$, then $f_0(x)=a\sin(x+\varphi)$ where $a$ and $\varphi$ are real constants. This result was extended to $\R^n$ by Strichartz \cite{Str} where $d/dx$ is substituted by the Laplacian on $\R^n$. While it is plausible to extend this theorem for other Riemannian manifolds or Lie groups, Strichartz showed that the result holds true for Heisenberg groups, but fails for hyperbolic 3-space. This negative result can be indeed extended to any Riemannian symmetric space of noncompact type. We observe that this failure is rooted in the $p$-dependance of the $L^p$-spectrum of the Laplacian on the hyperbolic spaces. Taking this into account we shall prove that for all rank one Riemannian symmetric spaces of noncompact type, or more generally for the harmonic $NA$ groups, the theorem actually holds true when uniform boundedness is replaced by uniform "almost $L^p$ boundedness". In addition we shall see that for the symmetric spaces this theorem is capable of characterizing the Poisson transforms of $L^p$ functions on the boundary, which some what resembles the original theorem of Roe on $\R$.Comment: 30 page

3D hand tracking.

The hand is often considered as one of the most natural and intuitive interaction modalities for human-to-human interaction. In human-computer interaction (HCI), proper 3D hand tracking is the first step in developing a more intuitive HCI system which can be used in applications such as gesture recognition, virtual object manipulation and gaming. However, accurate 3D hand tracking, remains a challenging problem due to the hand’s deformation, appearance similarity, high inter-finger occlusion and complex articulated motion. Further, 3D hand tracking is also interesting from a theoretical point of view as it deals with three major areas of computer vision- segmentation (of hand), detection (of hand parts), and tracking (of hand). This thesis proposes a region-based skin color detection technique, a model-based and an appearance-based 3D hand tracking techniques to bring the human-computer interaction applications one step closer. All techniques are briefly described below. Skin color provides a powerful cue for complex computer vision applications. Although skin color detection has been an active research area for decades, the mainstream technology is based on individual pixels. This thesis presents a new region-based technique for skin color detection which outperforms the current state-of-the-art pixel-based skin color detection technique on the popular Compaq dataset (Jones & Rehg 2002). The proposed technique achieves 91.17% true positive rate with 13.12% false negative rate on the Compaq dataset tested over approximately 14,000 web images. Hand tracking is not a trivial task as it requires tracking of 27 degreesof- freedom of hand. Hand deformation, self occlusion, appearance similarity and irregular motion are major problems that make 3D hand tracking a very challenging task. This thesis proposes a model-based 3D hand tracking technique, which is improved by using proposed depth-foreground-background ii feature, palm deformation module and context cue. However, the major problem of model-based techniques is, they are computationally expensive. This can be overcome by discriminative techniques as described below. Discriminative techniques (for example random forest) are good for hand part detection, however they fail due to sensor noise and high interfinger occlusion. Additionally, these techniques have difficulties in modelling kinematic or temporal constraints. Although model-based descriptive (for example Markov Random Field) or generative (for example Hidden Markov Model) techniques utilize kinematic and temporal constraints well, they are computationally expensive and hardly recover from tracking failure. This thesis presents a unified framework for 3D hand tracking, using the best of both methodologies, which out performs the current state-of-the-art 3D hand tracking techniques. The proposed 3D hand tracking techniques in this thesis can be used to extract accurate hand movement features and enable complex human machine interaction such as gaming and virtual object manipulation

Correspondence of butterfly and host plant diversity: Foundation for habitat restoration and conservation

At a spatial scale, the diversity of butterflies varies with numerous factors including the availability of the host plant species. In parity with this proposition, the correspondence of diversity of butterfly and plant in the background of the urban–rural gradient was evaluated using Kolkata, India, as a model study area. The results reveal significant positive correlation between the diversity of butterflies and the plants, with the different values for the suburban, rural, and urban areas. Identification of the butterfly loads for the plants in the respective areas can be useful in enhancing the conservation of the butterflies through enhanced plantation of the concerned plant species. Alternatively, the disclosure of the generalist and specialist pattern of the plant species preference by the butterflies may be useful in enhancing the population of the respective species in the concerned areas. The conservation strategy for butterfly species may be refined through the use of both or any one of the quantitative assessment of the butterfly–plant links in the urban–rural gradient in Kolkata, India, and similar places in the world