2D alpha-shapes to quantify retinal microvasculature morphology and their application to proliferative diabetic retinopathy characterisation in fundus photographs

The use of 2D alpha-shapes (α-shapes) to quantify morphological features of the retinal microvasculature could lead to imaging biomarkers for proliferative diabetic retinopathy (PDR). We tested our approach using the MESSIDOR dataset that consists of colour fundus photographs from 547 healthy individuals, 149 with mild diabetic retinopathy (DR), 239 with moderate DR, 199 pre-PDR and 53 PDR. The skeleton (centrelines) of the automatically segmented retinal vasculature was represented as an α-shape and the proposed parameters, complexity ($${Op\alpha }_{min}$$ O p α min ), spread (OpA), global shape (VS) and presence of abnormal angiogenesis (Gradα) were computed. In cross-sectional analysis, individuals with PDR had a lower $${Op\alpha }_{min}$$ O p α min , OpA and Gradα indicating a vasculature that is more complex, less spread (i.e. dense) and the presence of numerous small vessels. The results show that α-shape parameters characterise vascular abnormalities predictive of PDR (AUC 0.73; 95% CI [0.73 0.74]) and have therefore potential to reveal changes in retinal microvascular morphology.

Grid Partition Variable Step Alpha Shapes Algorithm

On the basis of Alpha Shapes boundary extraction algorithm for discrete point set, a grid partition variable step Alpha Shapes algorithm is proposed to deal with the shortcomings of the original Alpha Shapes algorithm in the processing of nonuniform distributed point set and multiconcave point set. Firstly, the grid partition and row-column index table are established for the point set, and the point set of boundary grid partition is quickly extracted. Then, the average distance of the k -nearest neighbors of the point is calculated as the value of α . For the point set of boundary grid partition extracted in the previous step, Alpha Shapes algorithm is used to quickly construct the point set boundary. The proposed algorithm is verified by experiments of simulated point set and measured point set, and it has high execution efficiency. Compared with similar algorithms, the larger the number of point sets is, the more obvious the execution efficiency is.

A step in the Delaunay mosaic of order k

Given a locally finite set $$X \subseteq {{\mathbb {R}}}^d$$ X ⊆ R d and an integer $$k \ge 0$$ k ≥ 0 , we consider the function $${\mathbf{w}_{k}^{}} :{\mathrm{Del}_{k}{({X})}} \rightarrow {{\mathbb {R}}}$$ w k : Del k ( X ) → R on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

Quantifying and characterising contemporary rockfall supply to a debris-covered glacier catchment

<p>Supraglacial debris extent and thickness is an important control on the ablation rate of a debris-covered glacier. Debris is supplied to the surface of a debris-covered glacier through several pathways with the primary source of this debris originating from rockfall in both the accumulation area, where debris is transported englacially downglacier, and the upper ablation area, where debris remains in the supraglacial environment while transported downglacier. Current quantification of debris supply to debris-covered glaciers is limited to headwall erosion rates determined through the dating of headwall derived supraglacial debris using <sup>10</sup>Be concentrations, or estimations of these rates using a ratio of supraglacial debris flux to the headwall catchment area. To increase the knowledge of the contemporary short-term estimations of these processes, repeat LiDAR scans of debris-contributing slopes were acquired during a single ablation season in both July and September at Miage Glacier, Italy. An area of ~7.7 km<sup>2</sup> comprising > 1.8 billion 3D points was scanned per survey epoch, covering ~33% of the glacierised area of Miage Glacier. Sequential scans were co-registered using an iterative closest point adjustment algorithm within CloudCompare. Manual filtering was used to remove snow, artefacts, and the glacier surface from the raw point clouds. To ease processing, the rock walls were segmented both horizontally and vertically within the catchment. Change detection was carried out using the M3C2 algorithm at a projection scale of 0.3 m and point clouds representing areas of significant change within the segment were obtained using a distributed 95<sup>th</sup> percentile confidence interval. The DBSCAN clustering algorithm was used to identify individual rockfall clusters, and the volume of each rockfall was calculated using both an iterative alpha-shapes approach. Finally, a bounding box approach was used to estimate the a, b and c axes and therefore shape of the individual rockfalls. Increasing the projection scale used within the M3C2 algorithm decreases the frequency of significant rockfalls found exponentially, and an iterative alpha shapes approach is the most computationally efficient volume estimation method. Our results show that the Miage Glacier catchment is dominated by small scale rockfall events, although at least one large-scale rockfall event is evident in the upper ablation area (validated by time-lapse imagery). This failure on a recently deglaciated area of rock wall highlights that slope response to glacial erosion can be rapid following periods of deglaciation.</p>

Single Ping Filtering of Multi-Beam Sounding Data Based on Alpha Shapes

From the standpoint of complex marine environments relative to underwater acoustic signal propagation, the application of multi-beam systems for the filtering and processing of multi-beam sounding data is critical. However, currently existing automatic filtering methods estimate data, which involve several calculations and require the respective computer to possess significant processing capabilities. In this study, raw data are measured using an interferometer multi-beam echo sounder, and the characteristics of the noise data are analyzed. The noise data are discontinuous; however, accurate data can be approximated as a continuous distribution. Under the assumption of continuous underwater terrain, in consideration of a circle of the appropriate radius rolling on the terrain profile, the continuous underwater terrain data can be extracted from the raw data by means of the alpha-shapes algorithm. Finally, on the basis of the measured data in a sail trial, the effectiveness of the proposed algorithm is verified.

Alpha-SIM: A quick 3D geometry model simplification approach to support aircraft EWIS routing

Routing design of aircraft Electrical Wiring Interconnection System (EWIS) is time-consuming and error-prone. A solution, which automatically routes the EWIS inside the aircraft Digital MockUp (DMU), has been proposed and presented in the previous publications. The DMU, however, includes over-detailed features, which hardly influence the routing results but significantly increase the geometry-involved computational time thus hampering any automated routing. These features cannot be easily and efficiently suppressed. Therefore, a quick 3 D geometry simplification method, named Alpha-SIM, is proposed to enable a quick simplification of the airframe components included in the DMU and improve the benefit of the aforementioned automatic EWIS routing approach. The method is inspired by Descriptive Geometry techniques and the 3 D modelling approach using 2 D sketches, and aims at removing very detailed and/or internal features while preserving the intuitive notional shape of the given CAD model. The intuitive notional shape is represented by a 3 D point cloud of the model outer boundary and their 2 D projections on user-defined planes. These 2 D projections are then processed such to generate a set of 2 D profiles, called Alpha-Shapes, which are used, eventually, to re-build the 3 D model of the DMU components in a simplified/de-featured manner. By controlling the density of the 3 D points and the Alpha value to generate the 2 D profiles from the point projections, various geometric approximation levels can be achieved. The results of the test cases demonstrate the efficiency and effectiveness of the proposed method on the geometry simplification for automatic EWIS routing.

Measurement of Particle Size of Loose Accumulation Based on Alpha Shapes (AS) and Hill Climbing-Region Growing (HC-RG) Algorithms

The loose accumulation CAUSED by landslide, collapse, debris flow, and mine blasting, exerts considerable negative influence to human activities. Besides, it can easily trigger secondary disaster under inner and outer geological conditions. Extraction and measurement of the particle of loose accumulation is of importance for prediction of slope stability and mine blasting. In this paper, the 3D laser scanning is utilized to collect the point clouds of granular materials in physical model (three types of materials) and landslide accumulation in field, respectively. Then, the alpha shapes (AS) and hill climbing-region growing (HC-RG) algorithms are introduced for identifying particles and finding their dimensions (e.g., particle number and radii). Comparison between the recognition results and reality shows that both algorithms can provide a good performance in laboratory physical model, and acceptable results can be obtained when applying two algorithm to field survey. AS algorithm needs less time to process data than HC-GR algorithm; however, the recognition from HC-RG algorithm is more accurate than that by AS algorithm.