Configuration products and quotients in geometric modeling

The six-dimensional space SE(3) is traditionally associated with the space of configurations of a rigid solid (a subset of Euclidean three-dimensional space R³). But a solid itself can be also considered to be a set of configurations, and therefore a subset of SE(3). This observation removes the artificial distinction between shapes and their configurations, and allows formulation and solution of a large class of problems in mechanical design and manufacturing. In particular, the configuration product of two subsets of configuration space is the set of all configurations obtained when one of the sets is transformed by all configurations of the other. The usual definitions of various sweeps, Minkowski sum, and other motion related operations are then realized as projections of the configuration product into R³. Similarly, the dual operation of configuration quotient subsumes the more common operations of unsweep and Minkowski difference. We identify the formal properties of these operations that are instrumental in formulating and solving both direct and inverse problems in computer aided design and manufacturing. Finally, we show that all required computations may be approximated using a fast parallel sampling method on a GPU and provide error estimates for the approximation.     

On downstream transmissions in EPON Protocol over Coax (EPoC): An analysis of Coax framing approaches and other relevant considerations

We present different mechanisms for downstream transmissions in the coax segment of Ethernet Passive Optical Network (EPON) Protocol over Coax (EPoC). EPoC is the transparent extension of EPON over a cable operator’s Hybrid Fiber-Coax network. For managing and controlling such a hybrid network, a network operator prefers to have a unified scheduling, management, and quality of service environment that includes both the optical and coax portions of the network. In EPoC, this is achieved by extending the EPON Medium Access Control to run over the coax physical layer, to have a centralized end-to-end network control from the cable head-end to the end users premises. In this paper, we focus on the downstream transmissions in EPoC. We study three different framing approaches for downstream coax frames based on how sub-carriers in an orthogonal frequency division multiplexed symbol are modulated. We discuss the merits and demerits of each approach and then compare them based on their control overheads and the maximum average data transmission rates each of them can achieve. We analyze how different parameters such as modulation profile, symbol duration, number of sub-carriers and length of resource blocks affect the data rates and the performance of downstream transmissions. We present simulation results to examine the implications of these factors on packet-level performance, such as delay. The results indicate that dynamic and hybrid framing approaches tend to perform better than static approaches, when traffic and usage pattern are identical to those in real-world scenarios. Finally, we outline the important engineering and research problems in this area which can be topics of future research.     

Non-commutative morphology: Shapes, filters, and convolutions

Group morphology is a generalization of mathematical morphology which makes an explicit distinction between shapes and filters. Shapes are modeled as point sets, for example binary images or 3D solid objects, while filters are collections of transformations (such as translations, rotations or scalings). The action of a filter on a shape generalizes the basic morphological operations of dilation and erosion. This shift in perspective allows us to compose filters independent of shapes, and leads to a non-commutative generalization of the Minkowski sum and difference which we call the Minkowski product and quotient respectively. We show that these operators are useful for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. To compute these new operators, we propose the use of group convolution algebras, which extend classical convolution and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.