Analysis and visualisation of movement: an interdisciplinary review

The processes that cause and influence movement are one of the main points of enquiry in movement ecology. However, ecology is not the only discipline interested in movement: a number of information sciences are specialising in analysis and visualisation of movement data. The recent explosion in availability and complexity of movement data has resulted in a call in ecology for new appropriate methods that would be able to take full advantage of the increasingly complex and growing data volume. One way in which this could be done is to form interdisciplinary collaborations between ecologists and experts from information sciences that analyse movement. In this paper we present an overview of new movement analysis and visualisation methodologies resulting from such an interdisciplinary research network: the European COST Action "MOVE - Knowledge Discovery from Moving Objects" (http://www.move-cost.info). This international network evolved over four years and brought together some 140 researchers from different disciplines: those that collect movement data (out of which the movement ecology was the largest represented group) and those that specialise in developing methods for analysis and visualisation of such data (represented in MOVE by computational geometry, geographic information science, visualisation and visual analytics). We present MOVE achievements and at the same time put them in ecological context by exploring relevant ecological themes to which MOVE studies do or potentially could contribute.

Mathematical model of a heterogeneous pulmonary acinus structure

The pulmonary acinus is a gas exchange unit distal to the terminal bronchioles. A model of its structure is important for the computational investigation of mechanical phenomena at the acinus level. We propose a mathematical model of a heterogeneous acinus structure composed of alveoli of irregular sizes, shapes, and locations. The alveoli coalesce into an intricately branched ductal tree, which meets the space-filling requirement of the acinus structure. Our model uses Voronoi tessellation to generate an assemblage of the alveolar or ductal airspace, and Delaunay tessellation and simulated annealing for the ductal tree structure. The modeling condition is based on average acinar and alveolar volume characteristics from published experimental information. By applying this modeling technique to the acinus of healthy mature rats, we demonstrate that the proposed acinus structure model reproduces the available experimental information. In the model, the shape and size of alveoli and the length, generation, tortuosity, and branching angle of the ductal paths are distributed in several ranges. This approach provides a platform for investigating the heterogeneous nature of the acinus structure and its relationship with mechanical phenomena at the acinus level.

Why behavioral neuroscience still needs diversity?: A curious case of a persistent need

In the past few decades, a substantial portion of neuroscience research has moved from studies conducted across a spectrum of animals to reliance on a few species. While this undoubtedly promotes consistency, in-depth analysis, and a better claim to unraveling molecular mechanisms, investing heavily in a subset of species also restricts the type of questions that can be asked, and impacts the generalizability of findings. A conspicuous body of literature has long advocated the need to expand the diversity of animal systems used in neuroscience research. Part of this need is utilitarian with respect to translation, but the remaining is the knowledge that historically, a diverse set of species were instrumental in obtaining transformative understanding. We argue that diversifying matters also because the current approach limits the scope of what can be discovered. Technological advancements are already bridging several practical gaps separating these two worlds. What remains is a wholehearted embrace by the community that has benefitted from past history. We suggest the time for it is now.

A classification and characterization of two-locus, pure, strict, epistatic models for simulation and detection

Background

The statistical genetics phenomenon of epistasis is widely acknowledged to confound disease etiology. In order to evaluate strategies for detecting these complex multi-locus disease associations, simulation studies are required. The development of the GAMETES software for the generation of complex genetic models, has provided the means to randomly generate an architecturally diverse population of epistatic models that are both pure and strict, i.e. all n loci, but no fewer, are predictive of phenotype. Previous theoretical work characterizing complex genetic models has yet to examine pure, strict, epistasis which should be the most challenging to detect. This study addresses three goals: (1) Classify and characterize pure, strict, two-locus epistatic models, (2) Investigate the effect of model ‘architecture’ on detection difficulty, and (3) Explore how adjusting GAMETES constraints influences diversity in the generated models.

Results

In this study we utilized a geometric approach to classify pure, strict, two-locus epistatic models by “shape”. In total, 33 unique shape symmetry classes were identified. Using a detection difficulty metric, we found that model shape was consistently a significant predictor of model detection difficulty. Additionally, after categorizing shape classes by the number of edges in their shape projections, we found that this edge number was also significantly predictive of detection difficulty. Analysis of constraints within GAMETES indicated that increasing model population size can expand model class coverage but does little to change the range of observed difficulty metric scores. A variable population prevalence significantly increased the range of observed difficulty metric scores and, for certain constraints, also improved model class coverage.

Conclusions

These analyses further our theoretical understanding of epistatic relationships and uncover guidelines for the effective generation of complex models using GAMETES. Specifically, (1) we have characterized 33 shape classes by edge number, detection difficulty, and observed frequency (2) our results support the claim that model architecture directly influences detection difficulty, and (3) we found that GAMETES will generate a maximally diverse set of models with a variable population prevalence and a larger model population size. However, a model population size as small as 1,000 is likely to be sufficient.

Art Forms in Nature: radiolaria from Haeckel and Blaschka to 3D nanotomography, quantitative image analysis, evolution, and contemporary art

The illustrations of the late nineteenth-/twentieth-century scientist/artist Ernst Haeckel, as depicted in his book Art Forms in Nature (originally in German as Kunstformen der Natur, 1898-1904), have been at the intersection of art, biology, and mathematics for over a century. Haeckel's images of radiolaria (microscopic protozoans described as amoeba in glass houses) have influenced various artists for over a century (glass artists Leopold and Rudolph Blaschka; sculptor Henry Moore; architects Rene Binet, Zaha Hadid, Antoni Gaudi, Chris Bosse and Frank Gehry; and designers-filmmakers Charles and Ray Eames). We focus on this history and extend the artistic, biological, and mathematical contributions of this interdisciplinary legacy by going beyond the 3D visual, topological, and geometric analyses of radiolaria to include the nanoscale with graph theory, spatial statistics, and computational geometry. We analyze multiple visualizations of radiolaria generated through Haeckel's images, light microscopy, scanning electron microscopy, micro- and nanotomography, and three-dimensional computer rendering. Mathematical analyses are conducted using the image analysis package "Ka-me: A Voronoi Image Analyzer." Further analyses utilize three-dimensional printing, laser etched crystalline glass art, and sculpture. Open sharing of three-dimensional nanotomography of radiolaria and other protozoa through MorphoSource enables new possibilities for artists, architects, paleontologists, structural morphologists, taxonomists, museum curators, and mathematical biologists. Distinctively, newer models of radiolaria fit into a larger context of productive interdisciplinary collaboration that continues Haeckel's legacy that lay a foundation for new work in biomimetic design and additive manufacturing where artistic and scientific models mutually and robustly generate wonder, beauty, utility, curiosity, insight, environmentalism, theory, and questions.

A Deep Learning-Based and Fully Automated Pipeline for Thoracic Aorta Geometric Analysis and Planning for Endovascular Repair from Computed Tomography

Feasibility assessment and planning of thoracic endovascular aortic repair (TEVAR) require computed tomography (CT)-based analysis of geometric aortic features to identify adequate landing zones (LZs) for endograft deployment. However, no consensus exists on how to take the necessary measurements from CT image data. We trained and applied a fully automated pipeline embedding a convolutional neural network (CNN), which feeds on 3D CT images to automatically segment the thoracic aorta, detects proximal landing zones (PLZs), and quantifies geometric features that are relevant for TEVAR planning. For 465 CT scans, the thoracic aorta and pulmonary arteries were manually segmented; 395 randomly selected scans with the corresponding ground truth segmentations were used to train a CNN with a 3D U-Net architecture. The remaining 70 scans were used for testing. The trained CNN was embedded within computational geometry processing pipeline which provides aortic metrics of interest for TEVAR planning. The resulting metrics included aortic arch centerline radius of curvature, proximal landing zones (PLZs) maximum diameters, angulation, and tortuosity. These parameters were statistically analyzed to compare standard arches vs. arches with a common origin of the innominate and left carotid artery (CILCA). The trained CNN yielded a mean Dice score of 0.95 and was able to generalize to 9 pathological cases of thoracic aortic aneurysm, providing accurate segmentations. CILCA arches were characterized by significantly greater angulation (p = 0.015) and tortuosity (p = 0.048) in PLZ 3 vs. standard arches. For both arch configurations, comparisons among PLZs revealed statistically significant differences in maximum zone diameters (p < 0.0001), angulation (p < 0.0001), and tortuosity (p < 0.0001). Our tool allows clinicians to obtain objective and repeatable PLZs mapping, and a range of automatically derived complex aortic metrics.

A simple algorithm for computing positively weighted straight skeletons of monotone polygons

We study the characteristics of straight skeletons of monotone polygonal chains and use them to devise an algorithm for computing positively weighted straight skeletons of monotone polygons. Our algorithm runs in [Formula: see text] time and [Formula: see text] space, where n denotes the number of vertices of the polygon.

Method for generating high-quality tetrahedral meshes of geological models by utilizing CGAL

High-quality computational meshes are crucial in the analysis of displacements and stabilities of rock and soil masses. In this paper, we present a method for generating high-quality tetrahedral meshes of geological models to be used in stability analyses of rock and soil masses. The method is implemented by utilizing the Computational Geometry Algorithms Library (CGAL). The input is a geological model consisting of triangulated surfaces, and the output is a high-quality tetrahedral mesh of the geological model. To demonstrate the effectiveness of the presented method, we apply it to generate a series of computational meshes of geological model, and we then analyse the stabilities of the rock and soil slopes on the basis of the generated tetrahedral mesh models. The applications demonstrate the effectiveness and practicability of the present method.•

A method for generating high-quality tetrahedral meshes of geological models is presented.

•

We evaluate the quality of the tetrahedral mesh of geological model using four metrics.

•

Three applications demonstrate the effectiveness and practicability of the presented method.

Novel Geometric Approach for Virtual Coiling

Endovascular coiling is a primary treatment for intra-cranial aneurysm, which deploys a thin and detachable metal wire inside the aneurysm so as to prevent its rupture. Emerging evidence from medical research and clinical practice has suggested that the coil configuration inside the aneurysm plays a vital role in properly treating aneurysm and predicting its outcome. In this paper, we propose a novel virtual coiling technique, called Ball Winding, for generating a coil configuration with ensured blocking ability. It can be used as an automatic tool for virtually simulating coiling before its implantation and thus optimizes such treatments. Our approach is based on integer linear programming and computational geometry techniques, and takes into consideration the packing density and coil distribution as the performance measurements. The resulting coiling is deployable (with the help of coil pre-shaping) and with minimized energy. Experimental results on both random and real aneurysm data suggest that our proposed method yields near optimal solution.

The n-dimensional k-vector and its application to orthogonal range searching

This work focuses on the definition and study of the n-dimensional k-vector, an algorithm devised to perform orthogonal range searching in static databases with multiple dimensions. The methodology first finds the order in which to search the dimensions, and then, performs the search using a modified projection method. In order to determine the dimension order, the algorithm uses the k-vector, a range searching technique for one dimension that identifies the number of elements contained in the searching range. Then, using this information, the algorithm predicts and selects the best approach to deal with each dimension. The algorithm has a worst case complexity of O ( n d ( k / n ) 2 / d ) , where k is the number of elements retrieved, n is the number of elements in the database, and d is the number of dimensions of the database. This work includes a detailed description of the methodology as well as a study of the algorithm performance.

A parallel solution to finding nodal neighbors in generic meshes

In this paper we specifically present a parallel solution to finding the one-ring neighboring nodes and elements for each vertex in generic meshes. The finding of nodal neighbors is computationally straightforward but expensive for large meshes. To improve the efficiency, the parallelism is adopted by utilizing the modern Graphics Processing Unit (GPU). The presented parallel solution is heavily dependent on the parallel sorting, scan, and reduction. Our parallel solution is efficient and easy to implement, but requires the allocation of large device memory.•Our parallel solution can generate the speedups of approximately 55 and 90 over the serial solution when finding the neighboring nodes and elements, respectively.•It is easy to implement due to the reason it does not need to perform the mesh-coloring before finding neighbors•There are no complex data structures, only integer arrays are needed, which makes our parallel solution very effective.

Mathematical modeling of pulmonary acinus structure: Verification of acinar shape effects on pathway structure using rat lungs

The pulmonary acinus is the gas exchange unit in the lung and has a very complex microstructure. The structure model is essential to understand the relationship between structural heterogeneity and mechanical phenomena at the acinus level with computational approaches. We propose an acinus structure model represented by a cluster of truncated octahedra in conical, double-conical, inverted conical, or chestnut-like conical confinement to accommodate recent experimental information of rodent acinar shapes. The basis of the model is the combined use of Voronoi and Delaunay tessellations and the optimization of the ductal tree assuming the number of alveoli and the mean path length as quantities related to gas exchange. Before applying the Voronoi tessellation, controlling the seed coordinates enables us to model acinus with arbitrary shapes. Depending on the acinar shape, the distribution of path length varies. The lengths are more widely spread for the cone acinus, with a bias toward higher values, while most of the lengths for the inverted cone acinus primarily take a similar value. Longer pathways have smaller tortuosity and more generations, and duct length per generation is almost constant irrespective of generation, which agrees well with available experimental data. The pathway structure of cone and chestnut-like cone acini is similar to the surface acini's features reported in experiments. According to space-filling requirements in the lung, other conical acini may also be acceptable. The mathematical acinus structure model with various conical shapes can be a platform for computational studies on regional differences in lung functions along the lung surface, underlying respiratory physiology and pathophysiology.

A Satisficing Framework for Environmental Policy Under Model Uncertainty

We propose a novel framework for the economic assessment of environmental policy. Our main point of departure from existing work is the adoption of a satisficing, as opposed to optimizing, modeling approach. Along these lines, we place primary emphasis on the extent to which different policies meet a set of goals at a specific future date instead of their performance vis-a-vis some intertemporal objective function. Consistent to the nature of environmental policymaking, our model takes explicit account of model uncertainty. To this end, the decision criterion we propose is an analog of the well-known success-probability criterion adapted to settings characterized by model uncertainty. We apply our criterion to the climate-change context and the probability distributions constructed by Drouet et al. (2015) linking carbon budgets to future consumption. Insights from computational geometry facilitate computations considerably and allow for the efficient application of the model in high-dimensional settings.

Maximum Matchings in Geometric Intersection Graphs

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O ( ρ 3 ω / 2 n ω / 2 ) time with high probability, where ρ is the density of the geometric objects and ω > 2 is a constant such that n × n matrices can be multiplied in O ( n ω ) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O ( n ω / 2 ) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [ 1 , Ψ ] can be found in O ( Ψ 6 log 11 n + Ψ 12 ω n ω / 2 ) time with high probability.

Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q . We present data structures of size O ( n · polylog n ) and with query time O ( ( k + 1 ) · polylog n ) time, where k is the number of reported pairs, for two classes of objects in R 2 : axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in  R 3 , we present a data structure of size O ( n n · polylog n ) and query time O ( ( n + k ) · polylog n ) . When the objects and query are fat, we obtain O ( ( k + 1 ) · polylog n ) query time using O ( n · polylog n ) storage.

Benchmark dataset for the Voronoi diagram of 3D spherical balls

In this paper, we present a dataset to be used for the construction of the Voronoi diagram of 3D spherical balls (VD-B3). The dataset consists of sphere arrangements including general, anomaly, and extreme cases. The dataset also includes protein models downloaded from RCSB Protein Data Bank (PDB). The dataset can be used as a standard benchmark dataset to verify and validate the correctness, efficiency, and robustness of the construction algorithm. The dataset is simple and easy to understand. The details of the experiment and analysis based on this dataset are presented in the original research article: "Robust Construction of the Voronoi Diagram of Spherical Balls in the Three-Dimensional Space" which introduces the topology-oriented incremental algorithm for the construction that is thoroughly validated and compared with two implementations of the well known edge-tracing algorithm.

EDIT Software: A tool for the semi-automatic 3D reconstruction of bladder cancer and urinary bladder of animal models

BACKGROUND AND OBJECTIVE

This study presents the EDIT software, a tool for the visualization of the urinary bladder anatomy in the 3D space and for its semi-automatic 3D reconstruction.

METHODS

The inner bladder wall was computed by applying a Region of Interest (ROI) feedback-based active contour algorithm on the ultrasound images while the outer bladder wall was calculated by expanding the inner borders to approach the vascularization area on the photoacoustic images. The validation strategy of the proposed software was divided into two processes. Initially, the 3D automated reconstruction was performed on 6 phantom objects of different volume in order to compare the software computed volumes of the models with the true volumes of phantoms. Secondly, the in-vivo 3D reconstruction of the urinary bladder for 10 animals with orthotopic bladder cancer, which range in different stages of tumor progression was performed.

RESULTS

The results showed that the minimum volume similarity of the proposed 3D reconstruction method applied on phantoms is 95.59%. It is noteworthy to mention that the EDIT software enables the user to reconstruct the 3D bladder wall with high precision, even if the bladder silhouette has been significantly deformed by the tumor. Indeed, by taking into account the dataset of the 2251 in-vivo ultrasound and photoacoustic images, the presented software performs segmentation with dice similarity 96.96% and 90.91% for the inner and the outer borders of the bladder wall, respectively.

CONCLUSIONS

This study delivers the EDIT software, a novel software tool that uses ultrasound and photoacoustic images to extract different 3D components of the bladder.

Few-Shot Learning Geometric Ensemble for Multi-label Classification of Chest X-Rays

This paper aims to identify uncommon cardiothoracic diseases and patterns on chest X-ray images. Training a machine learning model to classify rare diseases with multi-label indications is challenging without sufficient labeled training samples. Our model leverages the information from common diseases and adapts to perform on less common mentions. We propose to use multi-label few-shot learning (FSL) schemes including neighborhood component analysis loss, generating additional samples using distribution calibration and fine-tuning based on multi-label classification loss. We utilize the fact that the widely adopted nearest neighbor-based FSL schemes like ProtoNet are Voronoi diagrams in feature space. In our method, the Voronoi diagrams in the features space generated from multi-label schemes are combined into our geometric DeepVoro Multi-label ensemble. The improved performance in multi-label few-shot classification using the multi-label ensemble is demonstrated in our experiments (The code is publicly available at https://github.com/Saurabh7/Few-shot-learning-multilabel-cxray).

(Re)packing Equal Disks into Rectangle

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack n + k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0 . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0 . Our main algorithmic contribution is an algorithm that solves the repacking problem in time( h + k ) O ( h + k ) · | I | O ( 1 ) , where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

Rectangular partition for n-dimensional images with arbitrarily shaped rectilinear objects

Partitioning two- or multidimensional polygons into rectangular and rectilinear components is a fundamental problem in computational geometry. Rectangular and rectilinear decomposition have multiple applications in various fields of arts as well as sciences, especially when dissecting information into smaller chunks for efficient analysis, manipulation, identification, storage, and retrieval is essential. This article presents three simple yet elegant solutions for splitting geometric shapes (particularly non-diagonal ones) into non-overlapping and rectangular sub-objects. Experimental results suggest that each proposed method can successfully divide n-dimensional rectilinear shapes, including those with holes, into rectangular components containing no background elements. The proposed methods underwent testing on a dataset of 13 binary images, each with 1 … 4 dimensions, and the most extensive image contained 4096 elements. The test session consisted of 5 runs where starting points for decomposition were randomized where applicable. In the worst case, two of the three methods could complete the task in under 40 ms, while this value for the third method was around 11 s. The success rate for all the algorithms was 100 %.

New signal location method based on signal-range data for proximity tracing tools

Several technological solutions have emerged over the last several months to support proximity contact tracing to fight the COVID-19 pandemic. For this reason, today more than ever, accurate signal location is needed, even in indoor public areas (supermarkets, public transport, etc.). In a previous work, we proposed five methods to solve the problem of signal localization using elements of pole-polar geometry. The proposals were innovative, since they solved a geometric problem (locating a point in a coordinate system) only by applying concepts of geometry. Among these developed methods, the PPC (Pole-Polar Centroid model) was also presented. Although the PPC solves the problem of locating a device with better precision than conventional methods (based on numerical or optimization methods), its accuracy was found to be the worst among the five proposed geometric methods. In this context, this work proposes an extension to our PPC method, called the weighted Pole-Polar Centroid method (wPPC), which improves the accuracy of the previous PPC results. Such an extension does not change the complexity O(m ²) or the minimum dimensionality (m = 2) of nodes, which integrate a location network to perform the triangulation of such signals. Moreover, this extension estimates a device's location coordinates by means of the interaction, via signals, of this device with the network nodes distributed in any coordinate system. An IEEE 802.11 network infrastructure is used to accomplish the experiments. Errors in signal data are common, and our new proposed method, the wPPC, can mitigate the influence of these errors, produce more accurate results than the PPC, and outperform some of the other four proposed geometric methods and current numeric methods. Despite the use of an IEEE 802.11 network infrastructure for testing here, this range-based method for signal triangulation can be applied to any signal type (such as Wi-Fi, Bluetooth, and light and sound propagation).

Geometric uncertainty of patient-specific blood vessels and its impact on aortic hemodynamics: A computational study

In the context of numerical simulations of the vascular system, local geometric uncertainties have not yet been examined in sufficient detail due to model complexity and the associated large numerical effort. Such uncertainties are related to geometric modeling errors resulting from computed tomography imaging, segmentation and meshing. This work presents a methodology to systematically induce local modifications and perform a sufficient number of blood flow simulations to draw statistically relevant conclusions on the most commonly employed quantities of interest, such as flow rates or wall shear stress. The surface of a structured hexahedral mesh of a patient-specific aorta is perturbed by displacement maps defined via Gaussian random fields to stochastically model the local uncertainty of the boundary. Three different cases are studied, with the perturbation magnitude of 0.25, 0.5 and 1.0mm. Valid, locally perturbed meshes are constructed via an elasticity operator that extends surface perturbations into the interior. Otherwise, identical incompressible flow problems are solved on these meshes, taking physiological boundary conditions and Carreau fluid parameters into account. Roughly 300000 three-dimensional non-stationary blood flow simulations are performed for the three different perturbation cases to estimate the probability distributions of the quantities of interest. Convergence studies justify the spatial resolution of the employed meshes. Overall, the results suggest that moderate geometric perturbations result in reasonable engineering accuracy (relative errors in single-digit percentage range) of the quantities of interest, with higher sensitivity for gradient-related measures, noting that the observed errors are not negligible.

Automated landmarking of bends in vascular structures: a comparative study with application to the internal carotid artery

Automated tools for landmarking the internal carotid artery (ICA) bends have the potential for efficient and objective medical image-based morphometric analysis. The two existing algorithms rely on numerical approximations of curvature and torsion of the centerline. However, input parameters, original source code, comparability, and robustness of the algorithms remain unknown. To address the former two, we have re-implemented the algorithms, followed by sensitivity analyses. Of the input parameters, the centerline smoothing had the least impact resulting in 6-7 bends, which is anatomically realistic. In contrast, centerline resolution showed to completely over- and underestimated the number of bends varying from 3 to 33. Applying the algorithms to the same cohort revealed a variability that makes comparison of results between previous studies questionable. Assessment of robustness revealed how one algorithm is vulnerable to model smoothness and noise, but conceptually independent of application. In contrast, the other algorithm is robust and consistent, but with limited general applicability. In conclusion, both algorithms are equally valid albeit they produce vastly different results. We have provided a well-documented open-source implementation of the algorithms. Finally, we have successfully performed this study on the ICA, but application to other vascular regions should be performed with caution.

Breslow thickness: Geometric interpretation, potential pitfalls, and computer automated estimation

Breslow thickness is one of most important prognostic factor for cutaneous melanoma. To quantify the positions of the melanocytes, the Breslow thickness is defined on a distance metric that is reliable and easy to use in a clinical setting. In this letter, we want to highlight some pitfalls in this distance measurement arising from geometrical issues related to section bending and curling, and their consequences on computer automated estimation.

Connectivity with Uncertainty Regions Given as Line Segments

For a set Q of points in the plane and a real number δ ≥ 0 , let G δ ( Q ) be the graph defined on Q by connecting each pair of points at distance at most δ .We consider the connectivity of G δ ( Q ) in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set P of n - k points in the plane and a set S of k line segments in the plane, find the minimum δ ≥ 0 with the property that we can select one point p s ∈ s for each segment s ∈ S and the corresponding graph G δ ( P ∪ { p s ∣ s ∈ S } ) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in O ( f ( k ) n log n ) time, for a computable function f ( · ) . This implies that the problem is FPT when parameterized by k. The best previous algorithm uses O ( ( k ! ) k k k + 1 · n 2 k ) time and computes the solution up to fixed precision.