Objectives

• SNMP and MIBs, using NET-SNMP as an example, and using NETSNMP utilities to query MIB objects.

• Encryption, confidentiality, and authentication, including DES, RSA, MD5 and DSS.

• Application layer security, using SSH and Kerberos as examples.

• Transport layer security, including SSL and the secure Apache server.

• Network layer security, IPsec and Virtual Private Networks.

• Firewalls and IPTABLES.

• Accounting, auditing, and intrusion detection.

Network management

The Simple Network Management Protocol

In addition to configuring network devices when they are initially deployed, network management requires the performing of many tasks to run the network efficiently and reliably. A network administrator may need to collect statistics from a device to see if it is working properly, or monitor the network traffic load on the routers to see if the load is appropriately distributed. When there is a network failure, the administrator may need to go through the information collected from the nearby devices to identify the cause. The Simple Network Management Protocol (SNMP) is an application layer protocol for exchanging management information between network devices. It is the de facto network management standard in the Internet.

Figure 9.1 illustrates a typical SNMP management scenario, consisting of an SNMP manager and multiple managed devices. A managed device, e.g., a host computer or a router, maintains a number of Management Information Bases (MIB), which record local management related information.

Image formation is the task of constructing an image of a scene when given a set of noisy data that is dependent on that scene. Possibly some prior information about the scene is also given. Image formation also includes the task of refining a prior image when given additional fragmentary or degraded information about that image. Then the task may be called image restoration.

In the most fundamental problem of image restoration, one is given an image of a two-dimensional scene, but the detail of the image, in some way, is limited. For example, the image of the scene may be blurred or poorly resolved in various directions. Sophisticated signal-processing techniques, called deconvolution or deblurring, can enhance such an image. When the blurring function is not known but must be inferred from the image itself, these techniques are called blind deconvolution or blind deblurring. Problems of deconvolution are well known to be prone to computational instability, and great care is needed in the implementation of deconvolution algorithms.

Another task of image construction is estimating an image from partial knowledge of some of the properties of the image. An important instance of this task is estimating an image from the magnitude of its two-dimensional Fourier transform.

We may now conclude our overview of the “world according to (2-D) wavelets” [Bur98]. We have thoroughly analyzed the 2-D continuous wavelet transform, given some ideas about the discrete or discretized versions, discussed a large number of applications and generalizations (3-D, sphere, space–time). Where do we go now?

Why wavelets in the first place? When should one use them instead of other methods? Suppose we are facing a new signal or image. The very first question to ask is, what do we want to know or to measure from it? Depending on the answer, wavelets will or will not be useful. If we think they might be, we must next (i) choose a wavelet technique, discrete or continuous; (ii) then select a wavelet well adapted to the signal/image at hand, and (iii) determine the relevant parameter ranges. We emphasize that this approach is totally different from the standard one, based on Fourier methods. There is indeed no parameter to adjust here, the Fourier transform is universal. Wavelets on the other hand are extremely flexible, and the tool must be adapted each time to the situation at hand.

As for the first choice, discrete versus continuous WT, it is a fact that the vast majority of authors use the former, in particular if some data compression is required.

The 2-D CWT has been used by a number of authors, in a wide variety of problems [Com89,Mey91,Mey93]. In all cases, its main use is for the analysis of images, since image synthesis or compression problems are rather treated with the DWT. In particular, the CWT can be used for the detection or determination of specific features, such as a hierarchical structure, edges, filaments, contours, boundaries between areas of different luminosity, etc. Of course, the type of wavelet chosen depends on the precise aim. An isotropic wavelet (e.g. a Mexican hat) often suffices for pointwise analysis, but a directional wavelet (e.g. a Morlet or a conical wavelet) is necessary for the detection of oriented features in the signal. Somewhat surprisingly, a directional wavelet is often more efficient in the presence of noise.

In the next two chapters, we will review a number of such applications, including some nonlinear extensions of the CWT. First, in the present chapter, we consider various aspects of image processing. Then, in Chapter 5, we will turn to several fields of physics where the CWT has made an impact. Some of the applications are rather technical and use specific jargon. We apologize for that and refer the reader to the original papers for additional information.

Up to now, we have developed the 2-D CWT and a number of generalizations, relying in each case on the group-theoretical formalism. Given a class of finite energy signals and a group of transformations, including dilations, acting on them, one derives the corresponding continuous WT as soon as one can identify a square integrable representation of that group.

On the other hand, we have also briefly sketched the discrete WT and several transforms intermediate between the two. One conclusion of the study is that the pure DWT is too rigid, whereas redundancy is helpful, in that it increases both flexibility and robustness to noise of the transform. Indeed, the wavelet community has seen in the last few years a growing trend towards more redundancy and the development of tools more efficient than wavelets, such as ridgelets, curvelets, warplets, etc. The key word here is geometry: the new transforms and approximation methods take much better into account the geometrical features of the signals. To give a simple example, a smooth curve is in fact a 1-D object and it is a terrible waste (of times or bits) to represent it by a 2-D transform designed for genuine 2-D images.

It is therefore fitting to conclude the book by a chapter that covers these new developments.

What is wavelet analysis?

Wavelet analysis is a particular time- or space-scale representation of signals that has found a wide range of applications in physics, signal processing and applied mathematics in the last few years. In order to get a feeling for it and to understand its success, we consider first the case of one-dimensional signals. Actually the discussion in this introductory chapter is mostly qualitative. All the mathematically relevant properties will be described precisely and proved systematically in the next chapter for the two-dimensional case, which is the proper subject of this book.

It is a fact that most real life signals are nonstationary (that is, their statistical properties change with time) and they usually cover a wide range of frequencies. Many signals contain transient components, whose appearance and disappearance are physically very significant. Also, characteristic frequencies may drift in time (e.g., in geophysical time series – one calls them pseudo-frequencies). In addition, there is often a direct correlation between the characteristic frequency of a given segment of the signal and the time duration of that segment. Low frequency pieces tend to last for a long interval, whereas high frequencies occur in general for a short moment only. Human speech signals are typical in this respect: vowels have a relatively low mean frequency and last quite a long time, whereas consonants contain a wide spectrum, up to very high frequencies, especially in the attack, but they are very short.

In the previous chapter, we have discussed a number of applications of the 2-D CWT that belong essentially to the realm of image processing. Besides these, however, there are plenty of applications to genuine physical problems, in such diverse fields as astrophysics, geophysics, fluid dynamics or fractal analysis. Here the CWT appears as a new analysis tool, that often proves more efficient than traditional methods, which in fact rarely go beyond standard Fourier analysis. We will review some of these applications in the present chapter, without pretention of exhaustivity, of course. Our treatment will often be sketchy, but we have tried to provide always full references to the original papers.

Astronomy and astrophysics

Wavelets and astronomical images

Astronomical imaging has distinct characteristics. First, the Universe has a marked hierarchical structure, almost fractal. Nearby stars, galaxies, quasars, galaxy clusters and superclusters have very different sizes and live at very different distances, which means that the scale variable is essential and a multiscale analysis is in order, instead of the usual Fourier methods. This suggests wavelet analysis. Now, the main problem is that of detecting particular features, relations, groupings, etc., in images, which leads us to prefer the continuous WT over the discrete WT. Finally, there is in general no privileged direction, nor particular oriented features, in astrophysical images.

Wavelets are everywhere nowadays. Be it in signal or image processing, in astronomy, in fluid dynamics (turbulence), in condensed matter physics, wavelets have found applications in almost every corner of physics. In addition, wavelet methods have become standard in applied mathematics, numerical analysis, approximation theory, etc. It is hardly possible to attend a conference on any of these fields without encountering several contributions dealing with them. Correspondingly, hundreds of papers appear every year and new books on the topic get published at a sustained pace, with publishers strongly competing with each other. So, why bother to publish an additional one?

The answer lies in the finer distinction between various types of wavelet transforms. There is, indeed, a crucial difference between two approaches, namely, the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT). Furthermore, one has to distinguish between problems in one dimension (signal analysis) and problems in two dimensions (image processing), since the status of the literature is very different in the two cases.

Take first the one-dimensional case. Beginning with the classic textbook of Ingrid Daubechies [Dau92], several books, such as those of M. Holschneider [Hol95], B. Torrésani [Tor95] or A. Arnéodo et al. [Arn95], cover the continuous wavelet transform, in a more or less mathematically oriented approach.

Introduction

We live in a world where objects (cars, animals, men, birds, aeroplanes, the Sun, etc.) that surround us are constantly in relative motion. One would like to extract the motion information from the observation of the scene and use it for various purposes, such as detection, tracking and identification. In particular, tracking of multiple objects is of great importance in many real world scenarios. The examples include traffic monitoring, autonomous vehicle navigation, and tracking of ballistic missile warheads. Tracking is a complex problem, often requiring to estimate motion parameters – such as position, velocity – under very challenging situations. Algorithms of this type typically have difficulty in the presence of noise, when the object is obscured, in situations including crossing trajectories, and when highly maneuvering objects are present.

Most motion estimation (ME) techniques such as the ones based on block matching, optical flow, and phase difference [Jah97,280,281] assume that the object is constant from frame to frame. That is, the signature of the object does not change with time. Consequently, these techniques tend to have difficulty handling complex motion, particularly when noise is present.

The time-dependent continuous wavelet transform (CWT) is attractive as a tool for analysis, in that important motion parameters can be compactly and clearly represented.

In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ3, wavelets in ℝn (n > 3), and wavelets on the 2-sphere S2. Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis.

In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform.

Three-dimensional wavelets

Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].