25 research outputs found
Automatic creation of boundary-representation models from single line drawings
This thesis presents methods for the automatic creation of boundary-representation models of polyhedral objects from single line drawings depicting the objects. This topic is important in that automated interpretation of freehand sketches would remove a bottleneck in current engineering design methods. The thesis does not consider conversion of freehand sketches to line drawings or methods which require manual intervention or multiple drawings.
The thesis contains a number of novel contributions to the art of machine interpretation of line drawings. Line labelling has been extended by cataloguing the possible tetrahedral junctions and by development of heuristics aimed at selecting a preferred labelling from many possible. The ”bundling” method of grouping probably-parallel lines, and the use of feature detection to detect and classify hole loops, are both believed to be original. The junction-line-pair formalisation which translates the problem of depth estimation into a system of linear equations is new. Treating topological reconstruction as a tree-search is not only a new approach but tackles a problem which has not been fully investigated in previous work
Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels
In the 1820\u27s, Nikolai Ivanovich Lobachevski discovered and began to explore the world\u27s first non-Euclidean geometry. This crucial development in the history of mathematics was not recognized as such in his own lifetime. When his work finally found a sympathetic audience in the late 19th century, it was reinterpreted in the light of various intermediate developments (particularly Riemann\u27s conception of geometry), which were foreign to Lobachevski\u27s own way of thinking about the subject. Because our modern understanding of his work derives from these reinterpretations, many of Lobachevski\u27s most striking ideas have been forgotten. To recover them, I have produced an illuminated version of Lobachevski\u27s most accessible work, Geometrische Untersuchungen zur Theorie der Parallellinien (Geometric Investigations on the Theory of Parallels), a book that he published in 1840. I have produced a new English version of this work, together with extensive mathematical, historical, and philosophical commentary. The commentary expands and explains Lobachevski\u27s often cryptic statements and proofs, while linking the individual propositions of his treatise to the related work of his predecessors (including Gerolamo Saccheri, J.H. Lambert, and A.M. Legendre), his contemporaries (including J·nos Bolyai and Karl Friedrich Gauss), and his followers (including Eugenio Beltrami, Henri PoincarÈ, and David Hilbert). This dissertation supplies the contemporary reader with all of the tools necessary to unlock Lobachevski\u27s rich, beautiful, but generally inaccessible world
Word choice in mathematical practice: a case study in polyhedra
We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid's and Euler's word choices in their influential development and, in particular, their use of the term 'polyhedron'. Then, jumping to the 20th century, we look at word choices surrounding the use of the term `polyhedron' in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach between Grünbaum and Shephard on the one hand and that of Hilton and Pedersen on the other, elucidating that the conflict was engendered by disagreement over the proper conceptualization, and so also the appropriate word choices, in the study of polyhedra
From surfaces to objects : Recognizing objects using surface information and object models.
This thesis describes research on recognizing partially obscured objects using
surface information like Marr's 2D sketch ([MAR82]) and surface-based geometrical
object models. The goal of the recognition process is to produce a fully
instantiated object hypotheses, with either image evidence for each feature or
explanations for their absence, in terms of self or external occlusion.
The central point of the thesis is that using surface information should be
an important part of the image understanding process. This is because surfaces
are the features that directly link perception to the objects perceived (for
normal "camera-like" sensing) and because surfaces make explicit information
needed to understand and cope with some visual problems (e.g. obscured features).
Further, because surfaces are both the data and model primitive, detailed
recognition can be made both simpler and more complete.
Recognition input is a surface image, which represents surface orientation and
absolute depth. Segmentation criteria are proposed for forming surface patches
with constant curvature character, based on surface shape discontinuities which
become labeled segmentation- boundaries.
Partially obscured object surfaces are reconstructed using stronger surface based
constraints. Surfaces are grouped to form surface clusters, which are 3D
identity-independent solids that often correspond to model primitives. These are
used here as a context within which to select models and find all object features.
True three-dimensional properties of image boundaries, surfaces and surface
clusters are directly estimated using the surface data.
Models are invoked using a network formulation, where individual nodes
represent potential identities for image structures. The links between nodes are
defined by generic and structural relationships. They define indirect evidence relationships
for an identity. Direct evidence for the identities comes from the data
properties. A plausibility computation is defined according to the constraints inherent
in the evidence types. When a node acquires sufficient plausibility, the
model is invoked for the corresponding image structure.Objects are primarily represented using a surface-based geometrical model.
Assemblies are formed from subassemblies and surface primitives, which are
defined using surface shape and boundaries. Variable affixments between assemblies
allow flexibly connected objects.
The initial object reference frame is estimated from model-data surface relationships,
using correspondences suggested by invocation. With the reference
frame, back-facing, tangential, partially self-obscured, totally self-obscured and
fully visible image features are deduced. From these, the oriented model is used
for finding evidence for missing visible model features. IT no evidence is found,
the program attempts to find evidence to justify the features obscured by an unrelated
object. Structured objects are constructed using a hierarchical synthesis
process.
Fully completed hypotheses are verified using both existence and identity
constraints based on surface evidence.
Each of these processes is defined by its computational constraints and are
demonstrated on two test images. These test scenes are interesting because they
contain partially and fully obscured object features, a variety of surface and solid
types and flexibly connected objects. All modeled objects were fully identified
and analyzed to the level represented in their models and were also acceptably
spatially located.
Portions of this work have been reported elsewhere ([FIS83], [FIS85a], [FIS85b],
[FIS86]) by the author