Stéphane Marchand-Maillet

Discrete Convexity, Straightness, and the 16-Neighborhood

In this paper, we extend some results in discrete geometry based on the 8-neighborhood to that of the 16-neighborhood, which now includes the chessboard and the knight moves. We first present some analogies between an 8-digital arc and a 16-digital arc as represented by shortest paths on the grid. We present a transformation which uniquely maps a 16-digital arc onto an 8-digital arc (and vice versa). The grid-intersect-quantization (GIQ) of real arcs is defined with the 16-neighborhood. This enables us to define a 16-digital straight segment. We then present two new distance functions which satisfy the metric properties and describe the extended neighborhood space. Based on these functions, we present some new results regarding discrete convexity and 16-digital straightness. In particular, we demonstrate the convexity of a 16-digital straight segment. Moreover, we define a new property for characterizing a digital straight segment in the 16-neighborhood space. In comparison to the 8-neighborhood space, the proposed 16-neighborhood coding scheme offers a more compact representation without any loss of information.

A minimum spanning tree approach to line image analysis

We propose a graph theoretic approach for extracting the skeleton of a binary line image. Unlike other thinning methods, emphasis is placed on the preservation of the topology of both the foreground and the background. Such conditions guarantee a relevant resulting structure that can be used as input for pattern recognition. Using the underlying graph structure, we can readily formulate this problem as an optimisation problem. Local information such as centrality is given by a distance transform operation. Global information such as location of a branch end is given via a minimum weighted spanning tree which spans all foreground pixels. The resulting structure is then characterised as a union of central paths between end points with their adjacency inter-relationships. Other image characteristics (e.g. width and length of the branches) are also provided. Computational results applied on real images illustrate the noise insensitivity of this method.

Skeleton location and evaluation based on local digital width in ribbon-like images

In this paper, we present a model for characterising the skeleton of line images. For comparative purposes, we base our theoretical model formulation on the ideal ribbon-like non-branching image. The model developed is first contrasted with Blums model and the wave propagation model. The aim of the proposed model is to avoid spurious branches of the skeleton and to refine the definition of centrality at sharp angles of the input image. The underlying concept of this model is based on a new definition for local width, derived from the idea of minimum base segment. This is first introduced in the real space, and then discretised for characterising a discrete skeleton. A graph theoretic approach, which is model-independent, is introduced for locating skeletons of non-branching images. By applying the graph-theoretic algorithm, performance evaluation of the proposed model is contrasted with Blums original model by investigating reconstruction performance of the output skeletons.

Binary digital image characteristics

This chapter introduces the basis for the shape characterization of an image by detailing the common characteristics considered in image processing. It considers and studies the relationships of such characteristics with noise and presents definitions and algorithms in this context. In the model of a binary digital image, the two subsets of pixels are commonly defined within the image: the foreground (i.e., black pixels) and the background (i.e., white pixels). Binary digital image analysis is concerned with the characterization of properties in the set of foreground pixels. This chapter introduces the morphological study of the sets of discrete points representing pixels in a binary image. This work relies on the definition of connected components, which itself depends on the definition of a neighborhood for a pixel. The chapter concentrates on the neighborhoods defined on square lattices. It is apparent that extensions to other lattices are straight forward in most cases. Once components are characterized, the definition of morphological factors allows for their analysis at a global level. The classifications of components for further recognition processing can be achieved using such factors. The chapter also introduces the component-labeling problem. After such a processing step, each connected component is treated as a separate part of the image and forms the basic entity for morphological study. However, before initiating such a study, it may be necessary to remove redundant or unwanted information from a component. Such information is referred to as noise and methods for reducing it are presented.

Acquisition and storage

This chapter introduces the acquisition of image, the first step of any image processing application. The acquisition process is modeled as a mathematical operator, to control and quantify the quality of the approximation of continuous images by digital ones. The chapter presents and compares the data structures used in this context. Mapping from a continuous to a discrete image is the first step in any digital image processing application. Discrete data resulting from this digitization process is then, stored in a form, which is suitable for further processing. This chapter focuses on binary digital image processing and, hence, binary image acquisition. Data resulting from the acquisition process is typically composed of black and white pixels, represented by integer points on a lattice associated with 0–1 values. This chapter also presents the study of partitions, which models acquisition devices and their dual lattices. Different digitization methods are presented, and their usage is justified. The problem of image data storage is considered. Depending on the approach chosen for further processing, the storage varies, in order to facilitate access to data throughout the process. In this respect, different methods for image data storage are presented. The form in which image data is stored and the approach chosen for processing naturally define the type of data structures, which are to be used in this context. The chapter presents such data structures. It also introduces binary digital image data compression and presents its simple techniques.

Algorithmic graph theory

This chapter introduces and relates applied combinatorial mathematics and algorithmic graph theory in the context of digital image processing. New definitions on the analogies with digital image processing are given. Graph theory is an important mathematical approach that can be used for mapping complex problems onto simple representations and models. It is based on a robust mathematical background that allows for the definition of optimal solution techniques for such problems. Thus, efficient algorithms can be derived, which can solve a particular problem based on its graph representation. Graph theory is an area of research by itself and finds applications in many fields, such as operational research and theoretical computer science. It also finds applications in image processing, where the discrete nature of image representations makes its use consistent. It is often the case that graph theory is implicitly used when developing a solution to a particular problem related to image processing. The aim here is to relate image processing concepts to algorithmic graph theory in order to take an advantage of this approach for further developments. The chapter presents terminology and definitions used in graph theory, followed by the summarization of well-known algorithms, which exist for the solution of problems defined in the context of graph theory. The classes of equivalent problems are defined and solutions are presented for the abstract representations of each class. A number of algorithms are presented for the shortest path problem and the minimum weighted spanning tree problem. These results are related to digital image processing concepts.

Chapter 1 – Digital topology

Publisher Summary nThis chapter presents the construction of a consistent topology in the discrete space. It also provides robust basic definitions, such as connectivity and distance. Comparisons in relation to the continuous space are given and detailed throughout the study. Owing to the discrete nature of computers on which automated image processing is to be performed, images are typically given as sets of discrete points. To obtain a robust mathematical background for digital image processing, a formal study of such sets is to be developed. Then only, theoretical investigations can be carried out for presenting digital image processing operators. The acquisition step, whereby a continuous set is mapped onto a set of discrete points is introduced to characterize discrete sets, which represent binary digital images. Then, a topology is to be built in this context. This problem is addressed from the basis of neighborhood relationships to the definition of discrete sets. Based on the results derived, the construction of discrete distance functions is presented. Finally, the compatibility of such distance functions with Euclidean distance is studied. This last part also allows for the refinement of the definitions of discrete distance functions.

Chapter 8 – Some applications

Publisher Summary nThis chapter presents some of the practical applications of the different classes of binary digital images. It illustrates how the techniques presented throughout the book can be applied for performing image analysis. The use of different approaches is discussed in conjunction with the specific applications proposed. Five binary images are selected from different application classes. The chapter also illustrates the processing of a hand-drawn printed circuit image. The aim of this chapter is to convert this image into a vector form. It also presents the second application, where the input image represents a transportation network. The third image represents a fingerprint, where identification calls for the characterization of minutiae within the image. Different classification techniques for fingerprint images have been proposed that also rely on the global curvature of ridges within the fingerprint. The chapter describes the process of how thinning techniques can help in achieving this result. The text image presented illustrates the use of skeletonisation for hand-written text analysis. The idea is to define strokes and their interrelationship for recognition. The final application illustrates the processing of a generic image. The use of the skeleton for compression in relation to this type of image is discussed.

Chapter 2 – Discrete geometry

Publisher Summary nThis chapter presents geometrical definitions and the properties of discrete sets including straightness, convexity, and curvature. These definitions and properties are mostly drawn from Euclidean geometry. Consistency is maintained with the continuous space and comparisons with equivalent continuous concepts are given. Discrete geometry aims for a characterization of the geometrical properties of a set of discrete points. The geometrical properties of a set are understood to be global properties. In discrete geometry, points are grouped, thus, forming discrete objects, and it is the properties of these discrete objects that are under study. In contrast, digital topology allows for the study of the local properties between discrete points within such an object. Topological properties, such as connectivity and neighborhood are first used to define discrete objects and discrete geometry, then characterizes the properties of these discrete objects. A possible approach is to consider discrete objects as digitization of continuous objects and to map the properties of the original continuous object in the classic Euclidean geometry onto properties in discrete geometry. In this context, there is first a need for the definitions and models of digitization schemes. The resulting discrete geometrical properties will clearly be highly dependent on the digitization scheme. Discrete convexity is presented as a natural extension of discrete straightness. On the square lattice, the different definitions of discrete convexity have been given. It also details their interrelationships and presents discrete convex hulls based on these definitions.

Chapter 5 – Distance transformations

Publisher Summary nThis chapter presents distance transformations, a fundamental image analysis tool. The distance map resulting from this operation forms the representation of the image on which the rest of the processing will be undertaken. The chapter also presents and compares efficient algorithms for distance transformation involving different approaches. It studies in depth the linkage between Euclidean and discrete distance transformation. Models for image representation generally involve minimum distances between the pixels and borders of components, pixels closer to a given set of pixels than to another set. In this context, it is often the case that measurements are operated between a point and a set rather than between points. Such an operation defines an important tool in digital image processing, the distance transformation (DT). By storing its result into the distance map (DM), the computational effort can be reduced and the global properties of the image can be characterized. The operation of calculating the distance map of an image is referred to as distance mapping. Distance mapping forms the preprocessing step in an image analysis process. In most cases, following distance mapping, its distance map for further processing represents the image. For example, operations, such as smoothing, merging, and thinning rely on the computation of the distance map of the image. In turn, the distance map itself depends on the definition of the distance used for computing it. Both discrete and Euclidean distances may be used to define discrete or Euclidean distance maps, respectively.