IN this chapter we will study some basic principles of the probabilistic method, a combinatorial tool with many applications in computer science. This method is a powerful tool for demonstrating the existence of combinatorial objects. We introduce the basic idea through several examples drawn from earlier chapters, and follow that by a detailed study of the maximum satisfiability (MAX-SAT) problem. We then introduce the notion of expanding graphs and apply the probabilistic method to demonstrate their existence. These graphs have powerful properties that prove useful in later chapters, and we illustrate these properties via an application to probability amplification.

In certain cases, the probabilistic method can actually be used to demonstrate the existence of algorithms, rather than merely combinatorial objects. We illustrate this by showing the existence of efficient non-uniform algorithms for the problem of oblivious routing. We then present a particular result, the Lovász Local Lemma, which underlies the successful application of the probabilistic method in a number of settings. We apply this lemma to the problem of finding a satisfying truth assignment in an instance of the SAT problem where each variable occurs in a bounded number of clauses. While the probabilistic method usually yields only randomized or non-uniform deterministic algorithms, there are cases where a technique called the method of conditional probabilities can be used to devise a uniform, deterministic algorithm; we conclude the chapter with an exposition of this method for derandomization.

IN this chapter we present some general bounds on the tail of the distribution of the sum of independent random variables, with some extensions to the case of dependent or correlated random variables. These bounds are derived via the use of moment generating functions and result in “Chernoff-type” or “exponential" tail bounds. These Chernoff bounds are applied to the analysis of algorithms for global wiring in chips and routing in parallel communications networks. For applications in which the random variables of interest cannot be modeled as sums of independent random variables, martingales are a powerful probabilistic tool for bounding the divergence of a random variable from its expected value. We introduce the concept of conditional expectation as a random variable, and use this to develop a simplified definition of martingales. Using measure theoretic ideas, we provide a more general description of martingales. Finally, we present an exponential tail bound for martingales and apply it to the analysis of an occupancy problem.

The Chernoff Bound

In Chapter 3 we initiated the study of techniques for bounding the probability that a random variable deviates far from its expectation. In this chapter we focus on techniques for obtaining considerably sharper bounds on such tail probabilities.

The random variables we will be most concerned with are sums of independent Bernoulli trials; for example, the outcomes of tosses of a coin. In designing and analyzing randomized algorithms in various settings, it is extremely useful to have an understanding of the behavior of this sum.

Summary

This thesis has developed a coherent framework for analysing image sequences based on the affine camera, and has demonstrated the practical feasibility of recovering 3D structure and motion in a bottom–up fashion, using “corner” features. New algorithms have been proposed to compute affine structure, and these have then been applied to the problems of clustering and view transfer. The theory of affine epipolar geometry has been derived and applied to outlier rejection and rigid motion estimation. Due consideration has been paid to error and noise models, with a χ2 test serving as a termination criterion for cluster growth and outlier detection, and confidence limits in the motion parameters facilitating Kalman filtering.

On a practical level, all the algorithms have been implemented and tested on a wide range of sequences. The use of n points and m frames has lead to enhanced noise immunity and has also simplified the algorithms in important ways, e.g. local coordinate frames are no longer needed to compute affine structure or rigid motion parameters. Finally, the use of 3D information without explicit depth has been illustrated in a working system (e.g. for transfer).

In summary, the affine camera has been shown to provide a solid foundation both for understanding structure and motion under parallel projection, and for devising reliable algorithms.

Future work

There are many interesting problems for future work to address. First, the CI space interpretation of the motion segmentation problem is that each independently moving object contributes a different 3D linear subspace.

Introduction

The first competence required of a motion analysis system is the accurate and robust measurement of image motion. This chapter addresses the problem of tracking independently–moving (and possibly non–rigid) objects in a long, monocular image sequence. “Corner features” are automatically identified in the images and tracked through successive frames, generating image trajectories. This system forms the low–level front–end of our architecture (cf. Figure 1.1), making reliable trajectory computation of the utmost importance, for these trajectories underpin all subsequent segmentation and motion estimation processes.

We build largely on the work of Wang and Brady [156, 157], and extend their successful corner–based stereo algorithm to the motion domain. Their key idea was to base correspondence on both similarity of local image structure and geometric proximity. There are, however, several ways in which motion correspondence is more complex than stereo correspondence [90]. For one thing, objects can change between temporal viewpoints in ways that they cannot between spatial viewpoints, e.g. their shape and reflectance can alter. For another, the epipolar constraint is no longer hard–wired by once–off calibration of a stereo–rig; motion induces variable epipolar geometry which has to be continuously updated (if the constraint is to be used). Furthermore, motion leads to arbitrarily long image sequences (instead of frame–pairs), which requires additional tracking machinery. The benefits are that temporal integration facilitates noise resistance, resolves ambiguities over time, and speeds up matching (via prediction).

Our framework has two parts: the matcher performs two–frame correspondence while the tracker maintains the multi-frame trajectories. Each corner is treated as an independent feature at this level (i.e. assigned an individual tracker as in [26]), and is tracked purely within the image plane. Section 2.2 justifies this feature–based approach and establishes the utility of corners as correspondence tokens.

Introduction

This chapter tackles the motion estimation problem, using affine epipolar geometry as the tool. Given m distinct views of n points located on a rigid object, the task is to compute its 3D motion without any prior 3D knowledge. There are several reasons why many existing point–based motion algorithms are of limited practical use: the inevitable presence of noise is often ignored; unreasonable demands are often made on prior processing (e.g. a suitable perceptual frame must first be selected, the features must appear in every frame, etc.); algorithms often only work in special cases (e.g. rotation about a fixed axis); and some algorithms require batch processing, rather than more natural sequential processing.

Although the epipolar constraint has been widely used in perspective and projective motion applications [43, 57, 87] (e.g. to aid correspondence, recover the translation direction and compute rigid motion parameters), it has seldom been used under affine viewing conditions (though see [66, 79]). This chapter therefore makes the following contributions:

• Affine epipolar geometry is related to the rigid motion parameters, and Koenderink and van Doom's novel motion representation is formalised [79]. The scale, cyclotorsion angle and projected axis of rotation are then computed directly from the epipolar geometry (i.e. using two views). The only camera calibration parameter needed here is aspect ratio. A suitable error model is also derived.

• Images are processed in successive pairs of frames, facilitating extension to the m-view case in a sequential (rather than batch) processing mode.

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