12 results in Geometric Methods in Signal and Image Analysis
1 - Introduction
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Summary
This chapter provides a brief summary of what geometric methods for signal and image analysis are all about. Several applications to imaging, computer graphics, and sensor networks are discussed and illustrated. The diversified nature of these applications is powerful testimony to the practical usage of geometric methods.
What is signal and image analysis?
Signal and image analysis refers to the extraction of meaningful information from signals and images using digital signal and image processing techniques. In signal processing, for example, we measure, manipulate, or analyze information about a signal [1]. Such a signal is defined as a function of one or more independent variables that carries information. Examples of signals are daily high temperatures measured over a month, voltages and currents in a circuit, stock prices, our voices, music and speech, images, videos, and emails. Image processing, on the other hand, is the study of any algorithm that takes an image as input and returns an image as output, such as image enhancement, segmentation, compression, inpainting, and feature detection, to name just a few [2]. Application domains of image processing abound and include medical imaging, biology, satellite imagery, and biometrics. Other areas closely related to signal and image processing are computer vision and computer graphics.
Why geometric methods?
In light of the successful use of signal and image processing principles and methodologies in a broad range of areas of exceptional social and economic value, there is a growing demand from within academia and industry to devise robust computational methodologies that uncover the key geometric and topological information from signals and images. Geometry is a branch of mathematics concerned with rigid form, size, and location of objects [3], whereas topology is one of the younger fields of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, shrinking, and twisting but no tearing or gluing [4].
Frontmatter
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Contents
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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5 - Geometric and differential topology of manifolds
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Geometric topology focuses primarily on the study of manifolds, which are the high-dimensional analog of surfaces. The restriction to manifolds, as opposed to general topological spaces, allows for a more geometric structure than in general topology, which is the study of the axiomatic properties of topological spaces. Differential topology, on the other hand, is the study of differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. Unlike surfaces, manifolds in general need not lie in some ambient Euclidean space. Differentiable manifolds are used in almost all areas of engineering, ranging from signal/image processing and computer graphics to medical imaging and network optimization.
In this chapter, we introduce some of the basic terminology of geometric and differential topology. While appearing a bit more abstract than in the preceding chapters, our view of manifolds will be through the prism of applications, and will often invoke and exploit the local structure of a manifold which locally looks like a high-dimensional Euclidean space (i.e. a flat space which is often interpreted as a tangent space). Numerous books are devoted entirely to manifold theory (see [59–64] for complete treatments). With an applied goal in mind, we will keep our discussion on manifolds succinct, simple, and quite informal, by opting for a significant emphasis on applications of geometric and differential topology to imaging and computer graphics, while unveiling the relevant classical results on the theory of manifolds. We start by introducing the basic definitions of manifold theory in Section 5.1. In particular, we define and give illustrative examples of topological manifolds, smooth manifolds, smooth functions on manifolds, vector fields, pushforward and pullback, Whitney embedding theorem, connections on manifolds, and quotient topology. We then introduce in Section 5.2 the basic elements of Riemannian geometry, including Riemannian metric on a Riemannian manifold, the Laplace–Beltrami operator, and isometry between manifolds. In Section 5.3 we emphasize the relationship between the geometry and topology of a manifold through the topological invariants. We introduce the triangular mesh representation of manifolds.
References
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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2 - Fundamentals of group theory
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Motivated by the concept of symmetry, we present in this chapter the fundamental elements of group theory. Roughly speaking, symmetry is a transformation that leaves an object of study unchanged, meaning that the object looks the same from different points of view. The set of transformations that characterize the symmetry of an object naturally form a group. Group theory is a branch of mathematics that was inspired by these types of groups, and is of paramount importance in many areas of physics, chemistry, engineering, and computer science. In chemistry, for example, groups are used to classify crystal structures and the symmetries of molecules. In physics, groups are used for solving problems in atomic, molecular, and solid state physics. Groups are extensively used in cryptography, which is the science of encoding information so that only certain specified people can decode it. In addition, group theory has proven very useful in a wide variety of signal and image processing applications, including filter design, image edge detection, and deformable image registration and retrieval.
The outline of this chapter is as follows. In Section 2.1, we start by pointing out the interesting connection of groups with symmetry. Just as numbers can be used to measure size, groups can be used to measure symmetry. This relation of groups with symmetry reveals an important linkage between geometry and algebra. Then, we introduce the notion of a group, and describe in detail the group-theoretical concepts through illustrative examples. In particular, we take a detailed look at subgroups, cosets and normal subgroups, quotient groups, homomorphisms, cyclic and permutation groups. We also highlight matrix groups. Section 2.2 provides a very bare summary of some basic facts about topological spaces and metric spaces. We describe homeomorphisms between topological spaces and isometries between metric spaces, then highlight the connection between these two structure-preserving maps. Next, we introduce two important classes of groups, the topological and symmetry groups.
Dedication
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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6 - Computational algebraic topology
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Data analysis arises in numerous applications, and the solution to many tasks hinges upon the proximal structure in the data as is often encountered in graph-theoretic problems, or upon a model which itself is captured by the topological structure of the data. Thus far, we have discussed extensively the extent to which geometry could shed light on a given data analysis problem. In particular, the notion of metrics and their preservation under transformation is central to any information extraction and exploitation. It is well known, for instance, that the rigid transformation group preserves all distances between points, which can in turn be used in furthering the analysis. It may, for instance, be useful in comparing two objects which are indistinguishable up to a rotation. Topology, on the other hand, heeding no attention to metrics, and as further discussed below, keys on the characteristic features which remain invariant under a fairly large class of transformations. This is specifically the case for homomorphisms acting on objects, while incurring no tearing and/or no gluing of any associated points.
Intuitively, topology of an object describes its coarse features, or in other words its “shape” at a macro scale. Typical questions answered by topology about an object include: Is the object one connected piece? Does it have a hole? Does it have a void? The last notions, with a little thought, extend to any dimension higher than three [131].
A simple illustration of a topological characterization via a homeomorphic property is shown in Figure 6.1 (left) of a surface sheet with a hole which is preserved by a stretching transformation as shown in Figure 6.1 (right). Such a property may also be used to topologically compare two distinct objects. On that basis, the letter “M” is topologically different from “B” but equivalent (or homeomorphic) to “N”. It may indeed be seen that “M” may be continuously modified to “N” through a continuous shortening of the right straight edge and down the wedge to the cusp. A subsequent vertical appropriate lengthening from the cusp yields the “N”.
Preface
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Summary
This book is about the use of modern geometric methods for signal and image analysis. It provides a comprehensive coverage of the subject from the basic principles to state-of-the-art concepts and applications. The objective is to give the reader a sound understanding of the major theoretical concepts and computational approaches for applying geometric techniques and methodologies in solving various problems that arise naturally in signal and image processing, computer graphics, computer-aided design, bioinformatics, and other disciplines. The emphasis throughout is on intuitive and application-driven arguments. All methods are illustrated by well-chosen examples and applications, and are selected from core areas of modern geometric and topological computing. Furthermore, the purpose is for the reader to become aware of some recent developments in this fast-growing field.
Audience
The book is intended as a comprehensive and concise reference for geometric and topological methods in signal and image processing. The topics covered in this book are essential for research in numerical geometry and computational algebraic topology, and desirable for students, researchers, and practitioners pursuing research in signal and image processing, computer vision, computer graphics, computer-aided design, and other related fields.
The content grew from notes developed for graduate and undergraduate courses in signal processing, image processing, and computer graphics given at North Carolina State University and Concordia University, primarily targeted at electrical engineering, computer science, and software engineering students.
Chapter organization and topics covered
This book abandons the classical definition–theorem–proof model, and instead heavily relies on effective computational techniques with concrete applications to image analysis, computer vision, geometry processing, and computer graphics. The pitfalls of including all the technical details at the expense of foregone physical intuition of many heavily mathematical texts are largely avoided. The first chapter presents a brief motivation behind geometric methods and their various applications in imaging and computer graphics. Chapters 2 and 3 lay the foundations for our coverage of geometry and topology, and are essential to the rest of the book. The remaining three chapters are, however, almost completely independent of each other.
Index
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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4 - Differential geometry of curves and surfaces
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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Differential geometry refers to the study of geometric properties using the basic tools of differential and integral calculus. Differentiation is primarily concerned with determining a given function's rate of change, while integration is essentially used to determine displacement. Differential geometry of curves and surfaces appears in a broad variety of engineering and computer science areas, including signal and image processing, computer vision, computer graphics, and computer-aided design [5,49–53]. In image processing, for example, geometric methods are used extensively in a bevy of imaging applications, particularly in image enhancement, segmentation, compression, watermarking, inpainting, and retrieval. Geometric signal processing, on the other hand, naturally applies the fundamental principles of signal processing on 3D surfaces or, more generally, on graphs with various applications ranging from 3D shape analysis and recognition to social networks, machine learning, and the smart grid. Designing curves and surfaces for computer-aided design applications is a major driving force for the automotive, aerospace, and architecture industries, and is also of paramount importance in delivering advanced visual effects not only in the film industry, but also in the fast-growing video game industry.
In this chapter, we begin our study of the local theory of curves and surfaces in the context of differential geometry. We focus primarily on studying the geometric properties of parametric curves and surfaces. In Section 4.1, we define and present through illustrative examples the essential geometric properties of plane curves as well as space curves. In particular, we define the tangent, speed, length, arc-length, curvature, and torsion of parametrized curves. We also present the Frenet–Serret apparatus, which essentially determines the local geometry of curves. Moreover, we briefly describe the fundamental theorem of curves, which basically states that a space curve is uniquely determined by its curvature and torsion up to a rigid motion. Section 4.2 is devoted to the basic concepts of local surface theory, including the tangent plane, vector fields, Gauss map, first and second fundamental forms, and surface curvatures. Finally, we present in Section 4.3 an application of differential geometry to image analysis.
3 - Vector spaces
- Hamid Krim, North Carolina State University, Abdessamad Ben Hamza, Concordia University, Montréal
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- Geometric Methods in Signal and Image Analysis
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This chapter introduces the concepts of vector spaces and linear mappings between such spaces. Vector spaces are akin to geometry and consist of vectors that may be added together and multiplied by scalars. We present the necessary foundations for understanding these abstract concepts and also for further study in numerous applications of signal and image processing. The remainder of this chapter is organized as follows. Section 3.1 provides a formal introduction to vector spaces and their important properties, along with many illustrative examples. In Section 3.2, we study linear operators that map the vectors in one vector space to those in another, while preserving the operations that give structure to these vector spaces. We discuss when two vector spaces are essentially the same or isomorphic, and explore the properties of two special subspaces, the kernel and range, associated with a linear operator. We show that the effect of a linear operator is equivalent to multiplication by the associated matrix. Then we discuss the eigenanalysis of linear operators and their associated matrices. Matrices are used extensively in almost all numerical mathematical computations, and can help solve complicated problems involving linear operators by simply performing matrix multiplications. We also introduce linear functionals that map a vector space to a field of scalars. Section 3.3 introduces inner product spaces, orthonormal sets and bases, and normed vector spaces. We present several types of linear operators that are especially important in signal and image processing, and then we examine some elementary properties of these operators and their associated matrices. In Section 3.4, we briefly define the concept of a topological vector space. The generalized eigenvalue problem is discussed in Section 3.5. In Section 3.6, the singular value decomposition of a matrix is described, followed by an application to image compression. Section 3.7 examines in detail the principal component analysis technique, along with an application to outlier detection in multivariate data.
Vector space theory
Generally speaking, a vector may geometrically be defined as an object that has both a magnitude and direction.