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Map construction construction methods automatically produce and/or update street map datasets using vehicle tracking data. Enabled by the ubiquitous generation of geo-referenced tracking data, there has been a recent surge in map construction algorithms coming from different computer science domains. A crosscomparison of the various algorithms is still very rare, since (i) algorithms and constructed maps are generally not publicly available and (ii) there is no standard approach to assess the result quality, given the lack of benchmark data and quantitative evaluation methods. This work represents a first comprehensive attempt to benchmark such map construction algorithms. We provide an evaluation and comparison of seven algorithms using four datasets and four different evaluation measures. In addition to this comprehensive comparison, we make our datasets, source code of map construction algorithms and evaluation measures publicly available on mapconstruction.org. This site has been established as a repository for map construction data and algorithms and we invite other researchers to contribute by uploading code and benchmark data supporting their contributions to map construction algorithms.

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We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.

We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in IR d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low density polygonal curves.

We revisit the problem of computing Fréchet distance between polygonal curves under L 1 , L 2 , and L ∞ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of n vertices each, a ε-approximation of their discrete Fréchet distance can be computed in roughly O(nκ 3 log n/ε 3 ) time in three dimensions, if one of the curves is κ-bounded. Previously, only a κ-approximation algorithm was known. If both curves are the socalled backbone curves, which are widely used to model protein backbones in molecular biology, we can ε-approximate their Fréchet distance in near linear time in two dimensions, and in roughly O(n 4/3 log nm) time in three dimensions. In the second part, we propose a pseudo-output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the number of switching cells, which is quadratic in the worst case, but tends to be much smaller in practice.

The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fréchet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e., the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called κ-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant κ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fréchet distance, as well.

Using primary cell culture to screen for changes in neuronal morphology requires specialized analysis software. We developed NeuronMetrics for semi-automated, quantitative analysis of two-dimensional (2D) images of fluorescently labeled cultured neurons. It skeletonizes the neuron image using two complementary image-processing techniques, capturing fine terminal neurites with high fidelity. An algorithm was devised to span wide gaps in the skeleton. NeuronMetrics uses a novel strategy based on geometric features called faces to extract a branch number estimate from complex arbors with numerous neurite-to-neurite contacts, without creating a precise, contact-free representation of the neurite arbor. It estimates total neurite length, branch number, primary neurite number, territory (the area of the convex polygon bounding the skeleton and cell body), and Polarity Index (a measure of neuronal polarity). These parameters provide fundamental information about the size and shape of neurite arbors, which are critical factors for neuronal function. NeuronMetrics streamlines optional manual tasks such as removing noise, isolating the largest primary neurite, and correcting length for self-fasciculating neurites. Numeric data are output in a single text file, readily imported into other applications for further analysis. Written as modules for ImageJ, NeuronMetrics provides practical analysis tools that are easy to use and support batch processing. Depending on the need for manual intervention, processing time for a batch of approximately 60 2D images is 1.0-2.5 h, from a folder of images to a table of numeric data. NeuronMetrics' output accelerates the quantitative detection of mutations and chemical compounds that alter neurite morphology in vitro, and will contribute to the use of cultured neurons for drug discovery.