Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing

Constructing medial axis transform of planar domains with curved boundaries

The paper describes an algorithm for generating an approximation of the medial axis transform (MAT) for planar objects with free form boundaries. The algorithm generates the MAT by a tracing technique that marches along the object boundary rather than the bisectors of the boundary entities. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction or branch points and the search for these branch points is more efficient than while tracing the bisectors. The algorithm works for multiply connected objects as well. Results of implementation are provided

Optimal design of aluminium extrusion dies using a novel geometry based approach.

Aluminium extrusion is a forming process used for manufacturing straight and long aluminium products. Among all aspects of the process, extrusion die design is the key issue for producing high-quality extrudates. The approaches to design extrusion dies can be broadly classified into three groups; trial and error, empirical based approach and numerical simulation based approaches. By using the first two methodologies, the quality of extrusion die designs are intrinsically and strongly linked with designers' experience and skill. As the required forms for extrusions become more complex, these two approaches becomes less useful. Besides, since the design knowledge is held by designers, it is more often a 'black art', and the personnel movement can influence the design work significantly. On the other hand, with the advent of computers and greatly enhanced computing capability, many new approaches have been introduced for designing extrusion dies in last few decades. However, even with the current computing power, the numerical simulation approach has its limitations, particular in time required and even accuracy. Extrusion process involves complex constitutive relationships and large deformation of material. To overcome the limitations posed by current available design approaches, a new geometry based methodology has been proposed in this thesis. The new methodology combines empirical design formulae, geometry reasoning technique and optimization algorithm together. The work originates from the earlier work done by Miles et al. [1, 2, 3, 4], and Armstrong and his colleagues [5, 6, 7, 8, 9, 10, 11, 12], In this research work, a new knowledge representation scheme is developed so that historical data can be easily gathered and reused. By using empirical bearing length design formulae with historical data, a new bearing length estimation approach is introduced so that new profiles can be designed based on past good designs. A novel die layout design approach has also been developed and validated. This new method uses bearing length estimation algorithms with maximum bearing length difference to give radial or fiat layout for single/multi-hole dies. By using medial axis transform, a set of new geometry reasoning algorithms have been studied. These algorithms give a general and robust way to analyze two-dimensional geometry shapes. A brand-new die profile categories have been proposed to avoid the drawbacks held by current classification. A new algorithm and a set of new classifying criteria have been introduced. Based on medial axis transform and geometry reasoning technique, extrusion die profiles can be classified into different category correctly and efficiently. This research work shows that all the proposed approaches give several feedback paths in extrusion die design process. Therefore, not only historical data can be reused for new designs, but it is also possible to acquire and represent design knowledge and to optimize the whole design process

Exploring 3D Shapes through Real Functions

This thesis lays in the context of research on representation, modelling and coding knowledge related to digital shapes, where by shape it is meant any individual object having a visual appareance which exists in some two-, three- or higher dimensional space. Digital shapes are digital representations of either physically existing or virtual objects that can be processed by computer applications. While the technological advances in terms of hardware and software have made available plenty of tools for using and interacting with the geometry of shapes, to manipulate and retrieve huge amount of data it is necessary to define methods able to effectively code them. In this thesis a conceptual model is proposed which represents a given 3D object through the coding of its salient features and defines an abstraction of the object, discarding irrelevant details. The approach is based on the shape descriptors defined with respect to real functions, which provide a very useful shape abstraction method for the analysis and structuring of the information contained in the discrete shape model. A distinctive feature of these shape descriptors is their capability of combining topological and geometrical information properties of the shape, giving an abstraction of the main shape features. To fully develop this conceptual model, both theoretical and computational aspects have been considered, related to the definition and the extension of the different shape descriptors to the computational domain. Main emphasis is devoted to the application of these shape descriptors in computational settings; to this aim we display a number of application domains that span from shape retrieval, to shape classification and to best view selection.Questa tesi si colloca nell\u27ambito di ricerca riguardante la rappresentazione, la modellazione e la codifica della conoscenza connessa a forme digitali, dove per forma si intende l\u27aspetto visuale di ogni oggetto che esiste in due, tre o pi? dimensioni. Le forme digitali sono rappresentazioni di oggetti sia reali che virtuali, che possono essere manipolate da un calcolatore. Lo sviluppo tecnologico degli ultimi anni in materia di hardware e software ha messo a disposizione una grande quantit? di strumenti per acquisire, rappresentare e processare la geometria degli oggetti; tuttavia per gestire questa grande mole di dati ? necessario sviluppare metodi in grado di fornirne una codifica efficiente. In questa tesi si propone un modello concettuale che descrive un oggetto 3D attraverso la codifica delle caratteristiche salienti e ne definisce una bozza ad alto livello, tralasciando dettagli irrilevanti. Alla base di questo approccio ? l\u27utilizzo di descrittori basati su funzioni reali in quanto forniscono un\u27astrazione della forma molto utile per analizzare e strutturare l\u27informazione contenuta nel modello discreto della forma. Una peculiarit? di tali descrittori di forma ? la capacit? di combinare propriet? topologiche e geometriche consentendo di astrarne le principali caratteristiche. Per sviluppare questo modello concettuale, ? stato necessario considerare gli aspetti sia teorici che computazionali relativi alla definizione e all\u27estensione in ambito discreto di vari descrittori di forma. Particolare attenzione ? stata rivolta all\u27applicazione dei descrittori studiati in ambito computazionale; a questo scopo sono stati considerati numerosi contesti applicativi, che variano dal riconoscimento alla classificazione di forme, all\u27individuazione della posizione pi? significativa di un oggetto