Construction of trivariate compactly supported biorthogonal box spline wavelets

AbstractWe give a formula for the duals of the masks associated with trivariate box spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions. The biorthogonal wavelets may have arbitrarily high regularities

Reliable Solid Modelling Using Subdivision Surfaces

Les surfaces de subdivision fournissent une méthode alternative prometteuse dans la modélisation géométrique, et ont des avantages sur la représentation classique de trimmed-NURBS, en particulier dans la modélisation de surfaces lisses par morceaux. Dans ce mémoire, nous considérons le problème des opérations géométriques sur les surfaces de subdivision, avec l'exigence stricte de forme topologique correcte. Puisque ce problème peut être mal conditionné, nous proposons une approche pour la gestion de l'incertitude qui existe dans le calcul géométrique. Nous exigeons l'exactitude des informations topologiques lorsque l'on considère la nature de robustesse du problème des opérations géométriques sur les modèles de solides, et il devient clair que le problème peut être mal conditionné en présence de l'incertitude qui est omniprésente dans les données. Nous proposons donc une approche interactive de gestion de l'incertitude des opérations géométriques, dans le cadre d'un calcul basé sur la norme IEEE arithmétique et la modélisation en surfaces de subdivision. Un algorithme pour le problème planar-cut est alors présenté qui a comme but de satisfaire à l'exigence topologique mentionnée ci-dessus.Subdivision surfaces are a promising alternative method for geometric modelling, and have some important advantages over the classical representation of trimmed-NURBS, especially in modelling piecewise smooth surfaces. In this thesis, we consider the problem of geometric operations on subdivision surfaces with the strict requirement of correct topological form, and since this problem may be ill-conditioned, we propose an approach for managing uncertainty that exists inherently in geometric computation. We take into account the requirement of the correctness of topological information when considering the nature of robustness for the problem of geometric operations on solid models, and it becomes clear that the problem may be ill-conditioned in the presence of uncertainty that is ubiquitous in the data. Starting from this point, we propose an interactive approach of managing uncertainty of geometric operations, in the context of computation using the standard IEEE arithmetic and modelling using a subdivision-surface representation. An algorithm for the planar-cut problem is then presented, which has as its goal the satisfaction of the topological requirement mentioned above

Designing aesthetically pleasing freeform surfaces in a computer environment

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Architecture, February 2001.Includes bibliographical references (p. 151-160).Statement: If computational tools are to be employed in the aesthetic design of freeform surfaces, these tools must better reflect the ways in which creative designers conceive of and develop such shapes. In this thesis, I studied the design of aesthetically constrained freeform surfaces in architecture and industrial design, formulated a requirements list for a computational system that would aid in the creative design of such surfaces, and implemented a subset of the tools that would comprise such a system. This work documents the clay modeling process at BMW AG., Munich. The study of that process has led to a list of tools that would make freeform surface modeling possible in a computer environment. And finally, three tools from this system specification have been developed into a proof-of-concept system. Two of these tools are sweep modification tools and the third allows a user to modify a surface by sketching a shading pattern desired for the surface. The proof-of-concept tools were necessary in order to test the validity of the tools being presented and they have been used to create a number of example objects. The underlying surface representation is a variational expression which is minimized using the finite element method over an irregular triangulated mesh.by Evan P. Smyth.Ph.D

Machine learned boundary definitions for an expert's tracing assistant in image processing

Department Head: Anton Willem Bohm.Includes bibliographical references (pages 178-184).Most image processing work addressing boundary definition tasks embeds the assumption that an edge in an image corresponds to the boundary of interest in the world. In straightforward imagery this is true, however it is not always the case. There are images in which edges are indistinct or obscure, and these images can only be segmented by a human expert. The work in this dissertation addresses the range of imagery between the two extremes of those straightforward images and those requiring human guidance to appropriately segment. By freeing systems of a priori edge definitions and building in a mechanism to learn the boundary definitions needed, systems can do better and be more broadly applicable. This dissertation presents the construction of such a boundary-learning system and demonstrates the validity of this premise on real data. A framework was created for the task in which expert-provided boundary exemplars are used to create training data, which in turn are used by a neural network to learn the task and replicate the expert's boundary tracing behavior. This is the framework for the Expert's Tracing Assistant (ETA) system. For a representative set of nine structures in the Visible Human imagery, ETA was compared and contrasted to two state-of-the-art, user guided methods--Intelligent Scissors (IS) and Active Contour Models (ACM). Each method was used to define a boundary, and the distances between these boundaries and an expert's ground truth were compared. Across independent trials, there will be a natural variation in an expert's boundary tracing, and this degree of variation served as a benchmark against which these three methods were compared. For simple structural boundaries, all the methods were equivalent. However, in more difficult cases, ETA was shown to significantly better replicate the expert's boundary than either IS or ACM. In these cases, where the expert's judgement was most called into play to bound the structure, ACM and IS could not adapt to the boundary character used by the expert while ETA could