Objectives

• Network interfaces and interface configuration.

• Network load and statistics.

• The Address Resolution Protocol and its operations.

• ICMP messages and Ping.

• Concept of subnetting.

• Duplicate IP addresses and incorrect subnet masks.

Local area networks

Generally there are two types of networks: point-to-point networks or broadcast networks. A point-to-point network consists of two end hosts connected by a link, whereas in a broadcast network, a number of stations share a common transmission medium. Usually, a point-to-point network is used for long-distance connections, e.g., dialup connections and SONET/SDH links. Local area networks are almost all broadcast networks, e.g., Ethernet or wireless local area networks (LANs).

Point-to-Point networks

The Point-to-Point Protocol (PPP) is a data link protocol for PPP LANs. The main purpose of PPP is encapsulation and transmission of IP datagrams, or other network layer protocol data, over a serial link. Currently, most dial-up Internet access services are provided using PPP.

PPP consists of two types of protocols. The Link Control Protocol (LCP) of PPP is responsible for establishing, configuring, and negotiating the datalink connection, while for each network layer protocol supported by PPP, there is a Network Control Protocol (NCP). For example, the IP Control Protocol (IPCP) is used for transmitting IP datagrams over a PPP link. Once the link is successfully established, the network layer data, i.e., IP datagrams, are encapsulate in PPP frames and transmitted over the serial link.

The Fourier transform of a two-dimensional function — or of an n-dimensional function — can be defined by analogy with the Fourier transform of a one-dimensional function. A multidimensional Fourier transform is a mathematical concept. Because many engineering applications of the two-dimensional Fourier transform deal with two-dimensional images, it is common practice, and ours, to refer to the variables of a two-dimensional function as “spatial coordinates” and to the variables of its Fourier transform as “spatial frequencies.”

The study of the two-dimensional Fourier transform closely follows the study of the one-dimensional Fourier transform. As the study develops, however, the two-dimensional Fourier transform displays a richness beyond that of the one-dimensional Fourier transform.

The two-dimensional Fourier transform

A function, s(x, y), possibly complex, of two variables x and y is called a two-dimensional signal or a two-dimensional function (or, more correctly, a function of a two-dimensional variable). A common example is an image, such as a photographic image, wherein the variables x and y are the coordinates of the image, and s(x, y) is the amplitude. In a photographic image, the amplitude is a nonnegative real number. In other examples, the function s(x, y) may also take on negative or even complex values. Figure 3.1 shows a graphical representation of a two-dimensional complex signal in terms of the real and imaginary parts. Figure 3.2 shows the magnitude of the function depicted in two different ways: one as a three-dimensional graph, and one as a plan view.

As processing technology continues its rapid growth, it occasionally causes us to take a new point of view toward many long-established technological disciplines. It is now possible to record precisely such signals as ultrasonic or X-ray signals or electromagnetic signals in the radio and radar bands and, using advanced digital or optical processors, to process these records to extract information deeply buried within the signal. Such processing requires the development of algorithms of great precision and sophistication. Until recently such algorithms were often incompatible with most processing technologies, and so there was no real impetus to develop a general, unified theory of these algorithms. Consequently, it was scarcely noticed that a general theory might be developed, although some special problems were well studied. Now the time is ripe for a general theory of these algorithms. These are called algorithms for remote image formation, or algorithms for remote surveillance. This topic of image formation is a branch of the broad field of informatics.

Systems for remote image formation and surveillance have developed independently in diverse fields over the years. They are often very much driven by the kind of hardware that is used to sense the raw data or to do the processing.

Our immediate environment is a magnificent tapestry of information-bearing signals of many kinds: some are man-made signals and some are not, reaching us from many directions. Some signals, such as optical signals and acoustic signals, are immediately compatible with our senses. Other signals, as in the radio and radar bands, or as in the infrared, ultraviolet, and X-ray bands, are not directly compatible with human senses. To perceive one of these signals, we require a special apparatus to convert it into observable form.

A great variety of man-made sensors now exist that collect signals and process those signals to form some kind of image, normally a visual image, of an object or a scene of objects. We refer to these as sensors for remote surveillance. There are many kinds of sensors collected under this heading, differing in the size of the observed scene, as from microscopes to radio telescopes; in complexity, as from the simple lens to synthetic-aperture radars; and in the current state of development, as from photography to holography and tomography. Each of these devices collects raw sensor data and processes that data into imagery that is useful to a user. This processing might be done by a digital computer, an optical computer, or an analog computer. The development and description of the processing algorithms will often require a sophisticated mathematical formulation.

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.