The applications of Fortran span, very nearly, the whole period during which computers have been in general-purpose use. This is quite remarkable and, given the demise so many other high-level languages, it is quite difficult to know why. Possibly the original design concepts of John Backus – ease of use and efficiency of execution – have been two major factors. Another might be the devotion of Fortran's user community, who labor to keep it abreast of developments in programming techniques and to adapt it to ever-more demanding requirements.

Despite all the predictions, over several decades, that Fortran is about to become extinct, the language has shown itself to be remarkably resilient. Furthermore, over the last few years, it has been subject to new rounds of standardization, and the latest standard, Fortran 2008, should again extend the language's life. Against this background, it is very regrettable that old versions of Fortran live on, both in the form of antiquated courses given by incorrigible teachers and also as an outmoded concept in the minds of its detractors. After all, modern Fortran is a procedural, imperative, and compiled language with a syntax well suited to a direct representation of mathematical formulae. Its individual procedures may be compiled separately or grouped into modules, either way allowing the convenient construction of very large programs and procedure libraries. The language contains features for array processing, abstract data types, dynamic data structures, object-oriented programming, and parallel processing.

I was very pleased to be asked to write this foreword, having seen snapshots of the development of this book since its inception. I write this having just returned from BMVC 2011, where I found that others had seen draft copies, and where I heard comments like “What amazing figures!”, “It's so comprehensive!”, and “He's so Bayesian!”.

But I don't want you to read this book just because it has amazing figures and provides new insights into vision algorithms of every kind, or even because it's “Bayesian” (although more on that later). I want you to read it because it makes clear the most important distinction in computer vision research: the difference between “model” and “algorithm.” This is akin to the distinction that Marr made with his three-level computational theory, but Prince's two-level distinction is made beautifully clear by his use of the language of probability.

Why is this distinction so important? Well, let us look at one of the oldest and apparently easiest problems in vision: separating an image into “figure” and “ground.” It is still common to hear students new to vision address this problem just as the early vision researchers did, by reciting an algorithm: first I'll use PCA to find the dominant color axis, then I'll generate a grayscale image, then I'll threshold that at some value, then I'll clean up the holes using morphological operators.

This chapter concerns models for 2D and 3D shape. The motivation for shape models is twofold. First, we may wish to identify exactly which pixels in the scene belong to a given object. One approach to this segmentation problem, is to model the outer contour of the object (i.e., the shape) explicitly. Second, the shape may provide information about the identity or other characteristics of the object: it can be used as an intermediate representation for inferring higher-level properties.

Unfortunately, modeling the shape of an object is challenging; we must account for deformations of the object, the possible absence of some parts of the object and even changes in the object topology. Furthermore, the object may be partially occluded, making it difficult to relate the shape model to the observed data.

One possible approach to establishing 2D object shape is to use a bottom-up approach; here, a set of boundary fragments are identified using an edge detector (Section 13.2.1) and the goal is to connect these fragments to form a coherent object contour. Unfortunately, achieving this goal has proved surprisingly elusive. In practice, the edge detector finds extraneous edge fragments that are not part of the object contour and misses others that are part of the true contour. Hence it is difficult to connect the edge fragments in a way that correctly reconstructs the contour of an object.

In the final part of this book, we discuss four families of models. There is very little new theoretical material; these models are straight applications of the learning and inference techniques introduced in the first nine chapters. Nonetheless, this material addresses some of the most important machine vision applications: shape modeling, face recognition, tracking, and object recognition.

In Chapter 17 we discuss models that characterize the shape of objects. This is a useful goal in itself as knowledge of shape can help localize or segment an object. Furthermore, shape models can be used in combination with models for the RGB values to provide a more accurate generative account of the observed data.

In Chapter 18 we investigate models that distinguish between the identities of objects and the style in which they are observed; a prototypical example of this type of application would be face recognition. Here the goal is to build a generative model of the data that can separate critical information about identity from the irrelevant image changes due to pose, expression and lighting.

In Chapter 19 we discuss a family of models for tracking visual objects through time sequences. These are essentially graphical models based on chains such as those discussed in Chapter 11. However, there are two main differences. First, we focus here on the case where the unknown variable is continuous rather than discrete. Second, we do not usually have the benefit of observing the full sequence; we must make a decision at each time based on information from only the past.

The genesis of this great and beautiful book spans more than 20 years. It collects and unifies many theoretical notions and results published by Bruno Courcelle and others in a large number of articles.

The concept of a language to communicate with a computer, a machine or any kind of device performing operations is at the heart of Computer Science, a field that has truly thrived with the emergence of symbolic programming languages in the 1960s. Formalizing the algorithms that enable computers to calculate an intended result, to control a machine or a robot, to search and find the relevant information in response to a query, and even to imitate the human brain in actions such as measuring risk and making decisions, is the main activity of computer scientists as well as of ordinary computer users.

The languages designed for these tasks, which number by thousands, are defined in the first place by syntactic rules that construct sets of words and to which are then attached meanings. This understanding of a language was first conceived by structural linguists, in particular Nicolaï Troubetskoï, Roman Jacobson and Noam Chomsky, and has transformed Linguistics, the study of natural languages, by giving it new directions. It has also been extended to programming languages, which are artificial languages, and to the Lambda Calculus, one of many languages devised by logicians, among whom we can cite Kurt Gödel, Alonzo Church and Alan Turing, who aspired to standardize mathematical notation and to mechanize proofs.

This chapter presents the main definitions and results of this book and their significance, with the help of a few basic examples. It is written so as to be readable independently of the other chapters. Definitions are sometimes given informally, with simplified notation, and most proofs are omitted. All definitions will be repeated with the necessary technical details in subsequent chapters.

In Section 1.1, we present the notion of equational set of an algebra by using as examples a context-free language, the set of cographs and the set of series-parallel graphs. We also introduce our algebraic definition of derivation trees.

In Section 1.2, we introduce the notion of recognizability in a concrete way, in terms of properties that can be proved or refuted, for every element of the considered algebra, by an induction on any term that defines this element. We formulate a concrete version of the Filtering Theorem saying that the intersection of an equational set and a recognizable one is equational. It follows that one can decide if a property belonging to a finite inductive set of properties is valid for every element of a given equational set. We explain the relationship between recognizability and finite automata on terms.

In Section 1.3, we show with several key examples how monadic second-order sentences can express graph properties. We recall the fundamental equivalence of monadic second-order sentences and finite automata on words and terms.

As explained in the introduction of Section 1.7, there are no appropriate finite-state automata or finite-state transducers that work “directly” on graphs. Thus, for graphs, the role of automata and transducers is taken over by monadic second-order logic: instead of being accepted by an automaton, a set of graphs is defined by a monadic second-order sentence; and instead of being computed by a transducer, a graph transduction is defined by a definition scheme consisting of monadic second-order formulas. With respect to automata, the original motivation for this approach was Theorem 1.16 (cf. Theorem 5.82): for terms and words monadic second-order logic and finite-state automata have the same expressive power.

The aim of this chapter is to show that, in a certain sense, the automata-theoretic characterization of monadic-second order logic for sets of terms and words (Theorem 1.16) can be generalized to transductions. This means of course, that the automata should produce output, i.e., that they are transducers. We will concentrate on transductions that are partial functions, and in particular on deterministic devices, i.e., parameterless definition schemes and deterministic transducers. For these, we will show that monadic second-order transductions of words correspond to two-way finite-state transducers, and that monadic second-order transductions of terms correspond to (compositions of) tree-walking transducers. These transducers are well known from Formal Language Theory.

A two-way finite-state transducer is a finite-state automaton with a two-way read-only input tape and a one-way output tape. The input word is placed on the input tape between endmarkers, and the transducer has a reading head that is positioned over a cell of the input tape, at each moment of time.

We will define graph operations that generalize the concatenation of words. Some of these operations have been used in Section 1.1 to define the cographs and the series-parallel graphs. We will define actually two signatures of graph operations, hence two graph algebras. Both algebras will be defined in such a way that their equational sets are exactly the sets defined by certain context-free graph grammars. (The notion of an equational set has been presented informally in Section 1.1.4 and will be studied in detail in Chapter 3.)

Two main types of context-free sets of graphs defined by certain graph rewriting mechanisms have emerged from the intense research conducted from around 1980 and synthesized in the handbook [*Roz] edited by Rozenberg.

We will first define the HR graph algebra that corresponds in this respect to the hyperedge replacement grammars. It turns out that the operations of this algebra yield an exact characterization (as equational sets) of the sets of graphs of tree-width bounded by fixed integers. The terms built with the operations of the HR algebra can be seen as algebraic expressions of tree-decompositions. (Tree-width and the corresponding tree-decompositions are important for the construction of efficient graph algorithms and also for the characterization of the graphs that exclude a fixed graph as a minor.)

The second algebra, called the VR algebra, is defined so that its equational sets are those generated by the (context-free) vertex replacement grammars. A new graph complexity measure called clique-width has arisen in a natural way from the definition of the VR algebra, without having (yet) any independent combinatorial characterization.

This chapter defines monadic second-order logic, and shows how it can be used to formalize the expression of graph properties. Monadic second-order formulas can also be used to express the properties of sets of vertices and/or edges in graphs. The Recognizability Theorem (Section 5.3.8) implies that a class of graphs characterized as the class of (finite) models of a monadic second-order sentence is VR-recognizable, and that it is HR-recognizable (but not necessarily VR-recognizable) if the sentence is written with edge set quantifications.

It follows that the monadic second-order satisfiability problem for the class of graphs of tree-width or of clique-width at most k is decidable for each k. Applications to the construction of fixed-parameter tractable algorithms and other algorithmic consequences will be developed in Chapter 6.

Although our first applications concern graphs, we will give the definitions and prove the main results for the general case of relational structures because proofs are not more difficult, and the corresponding results apply in a uniform manner to labeled graphs represented, in several ways, by binary relational structures. However, relational structures with n-ary relations for n ≥ 2 are actually interesting by themselves. As we have shown in Section 1.9, they are useful to formalize betweenness and cyclic ordering (and for relational databases).

In Sections 5.1 and 5.2, we review basic notions of logic, define monadic second-order logic, and give some results that delineate its expressive power.