Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean\u2013hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean–hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Ground deformation and source geometry of the 30 October 2016 Mw 6.5 Norcia earthquake (Central Italy) investigated through seismological data, DInSAR measurements, and numerical modelling

We investigate the Mw 6.5 Norcia (Central Italy) earthquake by exploiting seismological data, DInSAR measurements, and a numerical modelling approach. In particular, we first retrieve the vertical component (uplift and subsidence) of the displacements affecting the hangingwall and the footwall blocks of the seismogenic faults identified, at depth, through the hypocenters distribution analysis. To do this, we combine the DInSAR measurements obtained from coseismic SAR data pairs collected by the ALOS-2 sensor from ascending and descending orbits. The achieved vertical deformation map displays three main deformation patterns: (i) a major subsidence that reaches the maximum value of about 98 cm near the epicentral zones nearby the town of Norcia; (ii) two smaller uplift lobes that affect both the hangingwall (reaching maximum values of about 14 cm) and the footwall blocks (reaching maximum values of about 10 cm). Starting from this evidence, we compute the rock volumes affected by uplift and subsidence phenomena, highlighting that those involved by the retrieved subsidence are characterized by significantly higher deformation values than those affected by uplift (about 14 times). In order to provide a possible interpretation of this volumetric asymmetry, we extend our analysis by applying a 2D numerical modelling approach based on the finite element method, implemented in a structural-mechanic framework, and exploiting the available geological and seismological data, and the ground deformation measurements retrieved from the multi-orbit ALOS-2 DInSAR analysis. In this case, we consider two different scenarios: the first one based on a single SW-dipping fault, the latter on a main SW-dipping fault and an antithetic zone. In this context, the model characterized by the occurrence of an antithetic zone presents the retrieved best fit coseismic surface deformation pattern. This result allows us to interpret the subsidence and uplift phenomena caused by the Mw 6.5 Norcia earthquake as the result of the gravitational sliding of the hangingwall along the main fault plane and the frictional force acting in the opposite direction, consistently with the double couple fault plane mechanism

Reliability-based G1 Continuous Arc Spline Approximation

In this paper, we present an algorithm to approximate a set of data points with G1 continuous arcs, using points' covariance data. To the best of our knowledge, previous arc spline approximation approaches assumed that all data points contribute equally (i.e. have the same weights) during the approximation process. However, this assumption may cause serious instability in the algorithm, if the collected data contains outliers. To resolve this issue, a robust method for arc spline approximation is suggested in this work, assuming that the 2D covariance for each data point is given. Starting with the definition of models and parameters for single arc approximation, the framework is extended to multiple-arc approximation for general usage. Then the proposed algorithm is verified using generated noisy data and real-world collected data via vehicle experiment in Sejong City, South Korea.Comment: 42 pages, 19 figures, Submitted to Computer Aided Geometric Desig

Exploring the geomorphological adequacy of the landslide susceptibility maps: A test for different types of landslides in the Bidente river basin (northern Italy)

Landslide susceptibility modelling is a crucial tool for implementing effective strategies in landslide risk mitigation. A plethora of statistical methods is available for generating accurate prediction images; however, the reliability of these models in terms of geomorphological adequacy is often overlooked by scholars. This critical flaw may result in concealed prediction errors, undermining the trustworthiness of the obtained maps. A key aspect of evaluating the geomorphological soundness of these models lies in factor analysis, specifically considering the correlation of explanatory variables with the final susceptibility score rather than solely focusing on their impact on model accuracy. This study delves into research conducted in the Bidente river basin (Italy) that analyes results obtained from slide, flow, and complex susceptibility models using Weight of Evidence (WoE) and Multivariate Adaptive Regression Splines (MARS) statistical methods. The research critically examines each factor class's role in defining susceptibility scores for different landslide typologies. The comparison between susceptibility maps generated by WoE and MARS for each typology (slide = 0.78; flow = 0.85; complex: 0.79) (slide = 0.78; flow = 0.85; complex: 0.79)reveals good to excellent prediction skill, with MARS demonstrating a 5 % higher performance index. The study emphasises the importance of spatial relationships between variables and landslide occurrences, highlighting that individual classes of variables influence the final susceptibility score based on their combined role with other predictor classes. In particular, in this study, results highlight that lithotecnical and landform classification classes delimit the landslide domain, while topographic attributes (steepness, curvatures, SPI and TWI) modulate the score inside. The proposed approach offers insights into investigating the geomorphological adequacy of landslide prediction images, emphasising the significance of factor analysis in evaluating model reliability and uncovering potential errors in susceptibility maps

Towards interactive global illumination effects via sequential Monte Carlo adaptation

Journal ArticleThis paper presents a novel method that effectively combines both control variates and importance sampling in a sequential Monte Carlo context while handling general single-bounce global illumination effects. The radiance estimates computed during the rendering process are cached in an adaptive per-pixel structure that defines dynamic predicate functions for both variance reduction techniques and guarantees well-behaved PDFs, yielding continually increasing efficiencies thanks to a marginal computational overhead

Theory and applications of bijective barycentric mappings

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable to extend the concept of barycentric coordinates from triangles to polygons. Several variants of such generalized barycentric coordinates have been proposed in recent years. An important application of barycentric coordinates consists of barycentric mappings, which allow to naturally warp a source polygon to a corresponding target polygon, or more generally, to create mappings between closed curves or polyhedra. The principal practical application is image warping, which takes as input a control polygon drawn around an image and smoothly warps the image by moving the polygon vertices. A required property of image warping is to avoid fold-overs in the resulting image. The problem of fold-overs is a manifestation of a larger problem related to the lack of bijectivity of the barycentric mapping. Unfortunately, bijectivity of such barycentric mappings can only be guaranteed for the special case of warping between convex polygons or by triangulating the domain and hence renouncing smoothness. In fact, for any barycentric coordinates, it is always possible to construct a pair of polygons such that the barycentric mapping is not bijective. In the first part of this thesis we illustrate three methods to achieve bijective mappings. The first method is based on the intuition that, if two polygons are sufficiently close, then the mapping is close to the identity and hence bijective. This suggests to ``split'' the mapping into several intermediate mappings and to create a composite barycentric mapping which is guaranteed to be bijective between arbitrary polygons, polyhedra, or closed planar curves. We provide theoretical bounds on the bijectivity of the composite mapping related to the norm of the gradient of the coordinates. The fact that the bound depends on the gradient implies that these bounds exist only if the gradient of the coordinates is bounded. We focus on mean value coordinates and analyse the behaviour of their directional derivatives and gradient at the vertices of a polygon. The composition of barycentric mappings for closed planar curves leads to the problem of blending between two planar curves. We suggest to solve it by linearly interpolating the signed curvature and then reconstructing the intermediate curve from the interpolated curvature values. However, when both input curves are closed, this strategy can lead to open intermediate curves. We present a new algorithm for solving this problem, which finds the closed curve whose curvature is closest to the interpolated values. Our method relies on the definition of a suitable metric for measuring the distance between two planar curves and an appropriate discretization of the signed curvature functions. The second method to construct smooth bijective mappings with prescribed behaviour along the domain boundary exploits the properties of harmonic maps. These maps can be approximated in different ways, and we discuss their respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples. The last method relies on a reformulation of complex barycentric mappings, which allows us to modify the ``speed'' along the edges to create complex bijective mappings. We provide some initial results and an optimization procedure which creates complex bijective maps. In the second part we provide two main applications of bijective mapping. The first one is in the context of finite elements simulations, where the discretization of the computational domain plays a central role. In the standard discretization, the domain is triangulated with a mesh and its boundary is approximated by a polygon. We present an approach which combines parametric finite elements with smooth bijective mappings, leaving the choice of approximation spaces free. This approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parametrization domain to the complex computational domain, thus creating a curved mesh of the latter. The second application addresses the meshing problem and the possibility to solve finite element simulations on polygonal meshes. In this context we present several methods to discretize the bijective mapping to create polygonal and piece-wise polynomial meshes

Improved prediction accuracy for disease risk mapping using Gaussian process stacked generalization.

Maps of infectious disease-charting spatial variations in the force of infection, degree of endemicity and the burden on human health-provide an essential evidence base to support planning towards global health targets. Contemporary disease mapping efforts have embraced statistical modelling approaches to properly acknowledge uncertainties in both the available measurements and their spatial interpolation. The most common such approach is Gaussian process regression, a mathematical framework composed of two components: a mean function harnessing the predictive power of multiple independent variables, and a covariance function yielding spatio-temporal shrinkage against residual variation from the mean. Though many techniques have been developed to improve the flexibility and fitting of the covariance function, models for the mean function have typically been restricted to simple linear terms. For infectious diseases, known to be driven by complex interactions between environmental and socio-economic factors, improved modelling of the mean function can greatly boost predictive power. Here, we present an ensemble approach based on stacked generalization that allows for multiple nonlinear algorithmic mean functions to be jointly embedded within the Gaussian process framework. We apply this method to mapping Plasmodium falciparum prevalence data in sub-Saharan Africa and show that the generalized ensemble approach markedly outperforms any individual method

Semi-Automated DIRSIG Scene Modeling from 3D LIDAR and Passive Imaging Sources

The Digital Imaging and Remote Sensing Image Generation (DIRSIG) model is an established, first-principles based scene simulation tool that produces synthetic multispectral and hyperspectral images from the visible to long wave infrared (0.4 to 20 microns). Over the last few years, significant enhancements such as spectral polarimetric and active Light Detection and Ranging (LIDAR) models have also been incorporated into the software, providing an extremely powerful tool for algorithm testing and sensor evaluation. However, the extensive time required to create large-scale scenes has limited DIRSIG’s ability to generate scenes “on demand.” To date, scene generation has been a laborious, time-intensive process, as the terrain model, CAD objects and background maps have to be created and attributed manually. To shorten the time required for this process, we are initiating a research effort that aims to reduce the man-in-the-loop requirements for several aspects of synthetic hyperspectral scene construction. Through a fusion of 3D LIDAR data with passive imagery, we are working to semi-automate several of the required tasks in the DIRSIG scene creation process. Additionally, many of the remaining tasks will also realize a shortened implementation time through this application of multi-modal imagery. This paper reports on the progress made thus far in achieving these objectives

AutoGraff: towards a computational understanding of graffiti writing and related art forms.

The aim of this thesis is to develop a system that generates letters and pictures with a style that is immediately recognizable as graffiti art or calligraphy. The proposed system can be used similarly to, and in tight integration with, conventional computer-aided geometric design tools and can be used to generate synthetic graffiti content for urban environments in games and in movies, and to guide robotic or fabrication systems that can materialise the output of the system with physical drawing media. The thesis is divided into two main parts. The first part describes a set of stroke primitives, building blocks that can be combined to generate different designs that resemble graffiti or calligraphy. These primitives mimic the process typically used to design graffiti letters and exploit well known principles of motor control to model the way in which an artist moves when incrementally tracing stylised letter forms. The second part demonstrates how these stroke primitives can be automatically recovered from input geometry defined in vector form, such as the digitised traces of writing made by a user, or the glyph outlines in a font. This procedure converts the input geometry into a seed that can be transformed into a variety of calligraphic and graffiti stylisations, which depend on parametric variations of the strokes