Abstract. We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. In particular, given a mapping, we show how to embed two paths on an n × n grid, and two caterpillar graphs on a 3n × 3n grid. We show that it is not always possible to simultaneously embed three paths. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2 ) × O(n 2 ) grid.

We present a fully automatic pipeline to train 3D Morphable Models (3DMMs), with contributions in pose normalisation, dense correspondence using both shape and texture information, and high quality, high resolution texture mapping. We propose a dense correspondence system, combining a hierarchical parts-based template morphing framework in the shape channel and a refining optical flow in the texture channel. The texture map is generated using raw texture images from five views. We employ a pixelembedding method to maintain the texture map at the same high resolution as the raw texture images, rather than using per-vertex color maps. The high quality texture map is then used for statistical texture modelling. The Headspace dataset used for training includes demographic information about each subject, allowing for the construction of both global 3DMMs and models tailored for specific gender and age groups. We build both global craniofacial 3DMMs and demographic sub-population 3DMMs from more than 1200 distinct identities. To our knowledge, we present the first public 3DMM of the full human head in both shape and texture: the Liverpool-York Head Model. Furthermore, we analyse the 3DMMs in terms of a range of performance metrics. Our evaluations reveal that the training pipeline constructs state-of-the-art models.

We address the problem of constrained exploration of an unknown graph G = (V, E) from a given start node s with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges of the graph, for example, it cannot jump between nonadjacent nodes. In the tethered robot case, if the tether (rope) has length l, then the robot must remain within distance l from the start node s. In the second variation, a fuel tank of limited capacity forces the robot to return to s after traversing C edges. The efficiency of algorithms for both variations of the problem is measured by the number of edges traversed during the exploration. We present an algorithm for a tethered robot that explores the graph in (|E|) edge traversals. The problem of exploration using a robot with a limited fuel tank capacity can be solved with a simple reduction from the tethered robot case and also yields a (|E|) algorithm. This improves on the previous best-known bound of O(|E|+|V | log 2 |V |). Since the lower bound for the graph exploration problems is (|E|), our algorithm is optimal within a constant factor.

Metopic synostosis is thought to have an incidence of about 1 in 15,000 births. Traditionally, this makes it the third most frequent single-suture craniosynostosis, after scaphocephaly (1 in 4200-8500) and plagiocephaly (1 in 11,000). Our units have, independently from each other, noted a marked increase in the number of metopic synostosis over the recent years. This is a pan-European, retrospective epidemiological study on the number of cases with metopic synostosis born between January 1, 1997, and January 1, 2006. This number was compared to the prevalence of scaphocephaly, the most frequently seen craniosynostosis. In the 7 units, a total of 3240 craniosynostosis were seen from 1997 until 2006. Forty-one percent (n = 1344) of those were sagittal synostosis, and 23% (n = 756) were metopic synostosis. There was a significant increase of the absolute number as well as of the percentage of metopic synostosis over these years (regression analysis, P = 0.017, R2 = 0.578) as opposed to a nonsignificant increase in the percentage of sagittal synostosis (P > 0.05, R2 = 0.368). The most remarkable increase occurred around 2000-2001, with the average of metopics being 20.1% from 1997 to 2000 and 25.5% from 2001 to 2005 (independent t-test, P = 0.002). The sagittal synostosis showed a smaller and nonsignificant increase in the same years: from 39.9% in 1997-2000 leading up to 42.5% in 2001-2005 (independent t-test, P > 0.05). The number of metopic synostosis has significantly increased over the reviewed period in all of our units, both in absolute numbers as in comparison to the total number of craniosynostosis.

Given a set S of n points in lRd, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and "fat" (that is, has a bounded aspect ratio). While previous hierarchical data structures, such as k-d trees, quadtrees, octrees, fair-split trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines alI of these properties simultaneously. The BAR tree data structure has numerous applications ranging from solving several geometric searching problems in fixed dimensional space to aiding in the visualization of graphs and three-dimensional worlds.