Topology-Preserving Deep Image Segmentation
Segmentation algorithms are prone to make topological errors on fine-scale structures,
e.g., broken connections. We propose a novel method that learns to segment
with correct topology. In particular, we design a continuous-valued loss function
that enforces a segmentation to have the same topology as the ground truth, i.e.,
having the same Betti number. The proposed topology-preserving loss function
is differentiable and we incorporate it into end-to-end training of a deep neural
network. Our method achieves much better performance on the Betti number error,
which directly accounts for the topological correctness. It also performs superiorly
on other topology-relevant metrics, e.g., the Adjusted Rand Index and the Variation
of Information. We illustrate the effectiveness of the proposed method on a broad
spectrum of natural and biomedical datasets.
This book is concerned with results in graph theory in which linear algebra and
matrix theory play an important role. Although it is generally accepted that linear
algebra can be an important component in the study of graphs, traditionally, graph
theorists have remained by and large less than enthusiastic about using linear
algebra. The results discussed here are usually treated under algebraic graph theory,
as outlined in the classic books by Biggs [20] and by Godsil and Royle [39]. Our
emphasis on matrix techniques is even greater than what is found in these and
perhaps the subject matter discussed here might be termed linear algebraic graph
theory to highlight this aspect.
After recalling some matrix preliminaries in the Chap. 1, the next few chapters
outline the basic properties of some matrices associated with a graph. This is
followed by topics in graph theory such as regular graphs and algebraic connectivity.
Distance matrix of a tree and its generalized version for arbitrary graphs, the
resistance matrix, are treated in the next two chapters. The final chapters treat other
topics such as the Laplacian eigenvalues of threshold graphs, the positive definite
completion problem, and matrix games based on a graph.
We have kept the treatment at a fairly elementary level and resisted the temptation
of presenting up-to-date research work. Thus, several chapters in this book
may be viewed as an invitation to a vast area of vigorous current research. Only a
beginning is made here with the hope that it will entice the reader to explore further.
In the same vein, we often do not present the results in their full generality, but
present only a simpler version that captures the elegance of the result. Weighted
graphs are avoided, although most results presented here have weighted, and hence
more general, analogs.
The references for each chapter are listed at the end of the chapter. In addition, a
master bibliography is included. In a short note at the end of each chapter, we
indicate the primary references that we used. Often, we have given a different
treatment, as well as different proofs, of the results cited. We do not go into an
elaborate description of such differences.
This monograph by one of the world's leading vision researchers provides a thorough, mathematically rigorous exposition of a broad and vital area in computer vision: the problems and techniques related to three-dimensional (stereo) vision and motion. The emphasis is on using geometry to solve problems in stereo and motion, with examples from navigation and object recognition. Faugeras takes up such important problems in computer vision as projective geometry, camera calibration, edge detection, stereo vision (with many examples on real images), different kinds of representations and transformations (especially 3-D rotations), uncertainty and methods of addressing it, and object representation and recognition. His theoretical account is illustrated with the results of actual working programs.Three-Dimensional Computer Vision proposes solutions to problems arising from a specific robotics scenario in which a system must perceive and act. Moving about an unknown environment, the system has to avoid static and mobile obstacles, build models of objects and places in order to be able to recognize and locate them, and characterize its own motion and that of moving objects, by providing descriptions of the corresponding three-dimensional motions. The ideas generated, however, can be used indifferent settings, resulting in a general book on computer vision that reveals the fascinating relationship of three-dimensional geometry and the imaging process.
The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the
engineering, mathematical, and scientific communities with significant developments in
harmonic analysis, ranging from abstract harmonic analysis to basic applications. The
title of the series reflects the importance of applications and numerical implementation,
but richness and relevance of applications and implementation depend fundamentally
on the structure and depth of theoretical underpinnings. Thus, from our point of view,
the interleaving of theory and applications and their creative symbiotic evolution is
axiomatic.
Chapter 1, An Overview of Unreal Engine, covers introductory content about what a game
engine is, specifically for Unreal Engine 4 and its history. You will get an overview of the
features of Unreal Engine 4 and how it can help you to create a game.
Chapter 2, Creating Your First Level, explains how to create your first room using the Box
Brush, add materials to texture the walls/floor, and learn how to place static objects to
enhance the look of the room.
Chapter 3, Game Objects – More and Move, covers the structure of a simple object type,
known as Static Mesh, and how objects in Unreal interact with one another. This chapter
also introduces Blueprints, which is the graphical scripting of Unreal Engine 4.
Chapter 4, Material and Light, shows you how to customize your level in greater detail by
learning how to create your own basic custom Material and how to use simple lights to
light up the interior of the level.
Chapter 5, Animation and AI, covers how animation works in Unreal Engine and how to
implement simple AI in your game level.
Chapter 6, A Particle System and Sound, explains how to add visual and sound effects to
your level.
Chapter 7, Terrain and Cinematics, shows you how to add the final touches to your level
using terrain manipulation and cinematics.
The WebVR API is an amazing addition to any web developer’s toolkit. It allows you
to develop simulated environments using HTML5, CSS3, and JavaScript. Unlike with
devices such as Microsoft HoloLens and Oculus Rift, you do not need any special software
development kit (SDK) to develop VR applications or games.
This paper introduces the mathematical framework of conformal
geometric algebra (CGA) as a language for computer graphics
and computer vision. Specifically it discusses a new method for pose and
position interpolation based on CGA which firstly allows for existing
interpolation methods to be cleanly extended to pose and position interpolation,
but also allows for this to be extended to higher-dimension
spaces and all conformal transforms (including dilations). In addition,
we discuss a method of dealing with conics in CGA and the intersection
and reflections of rays with such conic surfaces. Possible applications for
these algorithms are also discussed.
