Complexity of projected images of convex subdivisions

AbstractLet S be a subdivision of Rd into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d − 1, we construct a subdivision whose projected image has Ω(n⌊(3d−2)/2⌋) complexity, which is tight when d ⩽ 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane

Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.Comment: 20 pages, 5 figure

Implicitization of curves and (hyper)surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive

In vivo functional and myeloarchitectonic mapping of human primary auditory areas

In contrast to vision, where retinotopic mapping alone can define areal borders, primary auditory areas such as A1 are best delineated by combining in vivo tonotopic mapping with postmortem cyto- or myeloarchitectonics from the same individual. We combined high-resolution (800 μm) quantitative T(1) mapping with phase-encoded tonotopic methods to map primary auditory areas (A1 and R) within the "auditory core" of human volunteers. We first quantitatively characterize the highly myelinated auditory core in terms of shape, area, cortical depth profile, and position, with our data showing considerable correspondence to postmortem myeloarchitectonic studies, both in cross-participant averages and in individuals. The core region contains two "mirror-image" tonotopic maps oriented along the same axis as observed in macaque and owl monkey. We suggest that these two maps within the core are the human analogs of primate auditory areas A1 and R. The core occupies a much smaller portion of tonotopically organized cortex on the superior temporal plane and gyrus than is generally supposed. The multimodal approach to defining the auditory core will facilitate investigations of structure-function relationships, comparative neuroanatomical studies, and promises new biomarkers for diagnosis and clinical studies

Perceptions of Building-layout Complexity

This poster presents an experiment on judgments of design complexity, based on two modes of stimuli: the layouts of corridor systems in buildings shown in plan view and movies of simulated walkthroughs. Randomly selected stimuli were presented to 166 subjects: ‘experts’ (architects or students currently enrolled on an architectural course) and ‘lay people’ (all others). The aims were to investigate whether there were differences between these two groups in terms of their judgments of building complexity, effects of modality of stimuli and if any environmental measures (geometric or complexity-based) correlated with the assessments. The results were, first, there are differences between the judgments of the experts and non-experts, second, the effect of modality was negligible for lay people but evident for the ‘experts’, third, the judgments of both groups correlated highly with a number of environmental measures

Multi-layer approach to motion planning in obstacle rich environment

A widespread use of robotic technology in civilian and military applications has generated a need for advanced motion planning algorithms that are real-time implementable. These algorithms are required to navigate autonomous vehicles through obstacle-rich environments. This research has led to the development of the multilayer trajectory generation approach. It is built on the principle of separation of concerns, which partitions a given problem into multiple independent layers, and addresses complexity that is inherent at each level. We partition the motion planning algorithm into a roadmap layer and an optimal control layer. At the roadmap layer, elements of computational geometry are used to process the obstacle rich environment and generate feasible sets. These are used by the optimal control layer to generate trajectories while satisfying dynamics of the vehicle. The roadmap layer ignores the dynamics of the system, and the optimal control layer ignores the complexity of the environment, thus achieving a separation of concern. This decomposition enables computationally tractable methods to be developed for addressing motion planning in complex environments. The approach is applied in known and unknown environments. The methodology developed in this thesis has been successfully applied to a 6 DOF planar robotic testbed. Simulation results suggest that the planner can generate trajectories that navigate through obstacles while satisfying dynamical constraints