In various applications of computer vision and imageprocessing, it is required to detect points in animage, which characterize the visual content of thescene in its neighborhood and are distinguishableeven in other imaging instances of the same scene.These points are called key points of an image andthey are characterized by the functionaldistributions, such as distribution of brightnessvalues or color values, around its neighborhood foran image. For example, in the monocular and stereocamera geometries, various analyses involvecomputations of transformation matrices such as,homography between two scenes, fundamental matrixbetween two images of the same scene in a stereoimaging setup, etc. These transformation matricesare computed using key points of the same scenepoint of a pair of images. The image points of thesame scene point in different images of the sceneare called points ofcorrespondence or corresponding points. Key points ofimages are good candidates to form such pairs ofcorresponding points between two images of the samescene. Hence detection and matching of key points ina pair of images are fundamental tasks for suchgeometric analysis.

Consider Fig. 4.1, where images of the same scene arecaptured from two different views. Though theregions of structures in the images visuallycorrespond to each other, it is difficult toprecisely define points of correspondences betweenthem. Even an image of a two-dimensional (2-D)scene, such as 2-D objects on a plane, may gothrough various kinds of transformations, likerotation, scale, shear, etc. It may be required tocompute this transformation among such a pair ofimages. This is also a common problem of imageregistration.

learning Outcomes

After reading this chapter, the reader will be able to

• Understand the meaning of three processes of heat flow: conduction, convection, and radiation

• Know about thermal conductivity, diffusivity, and steady-state condition of a thermal conductor

• Derive Fourier's one-dimensional heat flow equation and solve it in the steady state

• Derive the mathematical expression for the temperature distribution in a lagged bar

• Derive the amount of heat flow in a cylindrical and a spherical thermal conductor

• Solve numerical problems and multiple choice questions on the process of conduction of heat

6.1 Introduction

Heat is the thermal energy transferred between different substances that are maintained at different temperatures. This energy is always transferred from the hotter object (which is maintained at a higher temperature) to the colder one (which is maintained at a lower temperature). Heat is the energy arising due to the movement of atoms and molecules that are continuously moving around, hitting each other and other objects. This motion is faster for the molecules with a largeramount of energy than the molecules with a smaller amount of energy that causes the former to have more heat. Transfer of heat continues until both objects attain the same temperature or the same speed. This transfer of heat depends upon the nature of the material property determined by a parameter known as thermal conductivity or coefficient of thermal conduction. This parameter helps us to understand the concept of transfer of thermal energy from a hotter to a colder body, to differentiate various objects in terms of the thermal property, and to determine the amount of heat conducted from the hotter to the colder region of an object. The transfer of thermal energy occurs in several situations:

• When there exists a difference in temperature between an object and its surroundings,

• When there exists a difference in temperature between two objects in contact with each other, and

• When there exists a temperature gradient within the same object.

Introduction

Statistical mechanics bridges the gaps between the laws of thermodynamics and the internal structure of the matter. Some examples are as follows:

• 1. Assembly of atoms in gaseous or liquid helium.

• 2. Assembly of water molecules in solid, liquid, or vapor state.

• 3. Assembly of free electrons in metal.

The behavior of all these abovementioned assemblies is totally different in different phases. Therefore, it is most significant to relate the macroscopic behavior of the system to its microscopic structure.

In this mechanics, most probable behavior of assembly are studied instead of individual particle interactions or behavior.

The behavior of assembly that is repeated a maximum time is known as most probable behavior.

hase Space

Six coordinates can fully characterize the state of any system:

• 1. Three for describing the position x, y, z and three for momentum Px, Py, Pz.

• 2. The combined position and momentum space (x, y, z, Px, Py, Pz) is called phase space.

• 3. The momentum space represents the energy of state,

For a system of N particles, there exists 3N position coordinates and 3N momentum coordinates. A single particle in phase space is known as a phase point, and the space occupied by it is known as µ-space.

olume Element ofµ-Space

4. Consider a particle having the position and momentum coordinates in the range.

Introduction

Humans have had a lengthy history of understanding electricity and magnetism. The tangible characteristics of light have also been studied. But in contrast to optics, electricity and magnetism—now known as electromagnetics—have been believed to be governed by different physical laws. This makes sense because optical physics as it was previously understood by humans differs significantly from the physics of electricity and magnetism. For instance, the ancient Greeks and Asians were aware of lode stone between 600 and 400 BC. Since 200 BC, China has been using the compass. The Greeks described static energy as early as 400 BC. But these oddities had no real effect until the invention of telegraphy. The voltaic cell or galvanic cell was created by Luigi Galvani and Alesandro Volta in the late 1700s, which led to the development of telegraphy. It quickly became clear that information could be transmitted using just two wires attached to a voltaic cell. The development of telegraphy was therefore prompted by this potential by the early 1800s. To learn more about the characteristics of electricity and magnetism, Andre-Marie Ampere (1823) and Michael Faraday (1838) conducted tests. Ampere's law and Faraday's law are consequently called after them. In order to comprehend telegraphy better, Kirchhoff voltage and current rules were also established in 1845. The data transmission mechanism was not well comprehended despite these laws. The cause of the data transmission signal's distortion was unknown. The ideal signal would alternate between ones and zeros, but the digital signal quickly lost its shape along a data transmission line.

A1.1 Basic matrix operations

A1.1.1 Definition (matrix)

A matrix is an ordered rectangular array of numbers (or functions). The numbers (or functions) are called the elements or the entries of the matrix.

A1.1.2 Dimension/size of a matrix

A matrix having m rows and a columns is a matrix of dimension m x n.

CHANCE PERMEATES OUR physical and mental universe. While the role of chance in human lives has had a longer history, starting with the more authoritative influence of the nobility, the more rationally sound theory of probability and statistics has come into practice in diverse areas of science and engineering starting from the early to mid-twentieth century. Practical applications of statistical theories proliferated to such an extent in the previous century that the American government-sponsored RAND corporation published a 600-page book that wholly consisted of a random number table and a table of standard normal deviates. One of the primary objectives of this book was to enable a computer-simulated approximate solution of an exact but unsolvable problem by a procedure known as the Monte Carlo method devised by Fermi, von Neumann, and Ulam in the 1930s–40s.

Statistical methods are the mainstay of conducting modern scientific experiments. One such experimental paradigm is known as a randomized control trial, which is widely used in a variety of fields such as psychology, drug verification, testing the efficacy of vaccines, agricultural sciences, and demography. These statistical experiments require sophisticated sampling techniques in order to nullify experimental biases. With the explosion of information in the modern era, the need to develop advanced and accurate predictive capabilities has grown manifold. This has led to the emergence of modern artificial intelligence (AI) technologies. Further, climate change has become a reality of modern civilization. Accurate prediction of weather and climatic patterns relies on sophisticated AI and statistical techniques. It is impossible to think of a modern economy and social life without the influence and role of chance, and hence without the influence of technological interventions based on statistical principles. We must begin this journey by learning the foundational tenets of probability and statistics.

EMPIRICAL TECHNIQUES rely on abstracting meaning from observable phenomena by constructing relationships between different observations. This process of abstraction is facilitated by appropriate measurements (experiments), suitable organization of data generated by measurements, and, finally, rigorous analysis of the data. The latter is a functional exercise that synthesizes information (data) and theory (model) and enables prediction of hitherto unobserved phenomena.1 It is important to underscore that a good theory (model) that explains a certain phenomenon well by appealing to a set of laws and conditions is expected to be a good candidate for predicting the same using reliable data. For example, a good model for the weight of a normal human being is w = m * h, where w and h refer to weight and height of the person, and m can be set to unity if appropriate units are chosen. A rational explanation of such a formula for weight based on anatomical considerations is perhaps very reasonable. From an empirical standpoint, if we collect height and weight data of normal humans, we will notice that a linear model of the form w = m * h represents the data reasonably well and may be used to predict the weight of the person based on the height of the person. This fact ascertains a functional symmetry between explanation and prediction. Therefore, a good predictive model must automatically be able to explain the data (and related events) well.

Learning Outcomes

After reading this chapter, the reader will be able to

• Express the meaning of sphere of influence and collision frequency

• Derive the distribution function for the free paths among the molecules and demonstrate the concept of mean free path

• Calculate the expression for mean free path following Clausius and Maxwell

• Derive the expression for pressure exerted by a gas using the survival equation

• Calculate the expressions for viscosity, thermal conductivity, and diffusion coefficient of a gaseous system

• Demonstrate Brownian motion with its characteristics and calculate the mean square displacement of a particle executing Brownian motion

• State the idea of a random walk problem

• Solve numerical problems and multiple choice questions on the mean free path, viscosity, thermal conduction, diffusion, Brownian motion, and random walk

4.1 Introduction

The molecules of an ideal gas are considered as randomly moving point particles. From the concept of kinetic theory of gases (KTG), it is well established that even at room temperature, such point molecules of the ideal gas move at very large speeds. The average value of this speed can be determined assuming that the molecules obey Maxwell's speed distribution law and is given by the following expression

Color is a psycho-physiological property of humanvisual experiences when the eyes look at objects andlight. Color is not a physical property of thoseobjects or light, rather, it is the result of aninteraction between physical light in theenvironment and human visual system (Palmer, 1999).For processing color images, it is required todevelop an understanding on how colors arerepresented following human perception.

3.1 Light sources

A broad range of electromagnetic spectrum, shown inFig. 3.1, consists of electromagnetic waves rangingfrom very long wavelengths at radio waves to veryhigh frequency at gamma waves. A very narrowinterval in this spectrum, toward the higher end ofspectral frequencies, accounts for the visible raysand it is called the visiblespectrum. The light and colors that ahuman eye perceives relate to the frequencies ofwaves that fall under the visible spectrum. Apictorial representation of the correspondence ofwavelengths in the visible range of the spectrum todifferent perceived colors has been shown in Fig.3.1. There are seven distinguishable colors in thefigure, violet, indigo, blue, green, yellow, orange,and red, usually known in order of their increasingwavelengths by the acronym of VIBGYOR. The luminancesensitivity function that is shown as a curve inFig. 3.1 is a function of the wavelength. It isempirically observed that the sensitivity of thehuman visual system is maximum in the green zone ofthe visible spectrum. The luminance sensitivityfunction gradually decays toward violet (higherfrequencies) and red (lower frequencies) from thegreen zone, as shown in the figure by the whitecurve.

Operational Amplifiers: Introduction

An operational amplifier (op-amp) is a very prominent active device used in analog integrated circuit (IC) design. Prominence is due to the widespread and diverse areas of applications of the op-amps as its parameters are very close to ideal in a certain range of operating frequencies. Apart from basic arithmetic operations such as addition, multiplication, and integration, op-amps are also widely employed as amplifiers, wave shaping circuits, active filters, log/anti-logarithmic amplifiers, nonlinear function generators, and in analog-to digital and digital-to-analog conversion, and so on.

Figure 8.1(a) shows a pin connection diagram of the most commonly used type-741 op-amp; it needs a dual power supply, has two terminals for inverting and non-inverting inputs, one terminal for the output, and three terminals without any connections for simple applications. Dual op-amps and quad op-amp ICs with matching characteristics are also available.

Op-amp is essentially a high-gain differential amplifier (DA) that can be shown in its simplest form as represented in Figure 8.1(b). The output voltage of the op-amp is the difference between the two input voltages multiplied by the high-gain factor A, so the output voltage is expressed as:

The differential gain A is frequency dependent in a practical op-amp. Therefore, as a first approximation, it is represented by a single-pole roll-off model given below.

Key Concepts

• • The growing share of electricity in the energy sector

• • The connection of electricity and global warming

• • Important terms related to electricity

• • Conventional sources of electricity generation

• • Green and renewable sources of electricity generation

• • Smart grid

Introduction

Electricity is the fundamental driver for growth of the modern society. The availability of reliable electric supply is a priority for any residential, industrial, or commercial setup. With the rapid proliferation of digital appliances and the critical role they are playing in our daily life, the dependence on high-quality electric power supply has further increased manifold.

Electricity started as a source of energy for lighting, replacing oil and gas-based lamps. But at that time very few people would have realized that slowly this new source of energy will ‘capture’ the whole residential, industrial, and workplace setup. It is difficult to imagine our lives without electricity now – starting from heating our meals, washing and drying of our clothes, heating the water, keeping the house or office cool or hot to running all kinds of entertainment and communication appliances. This source of energy has turned into an omnipresent phenomenon in our lives. Electricity is the main driver behind technologies related to the Internet and communication also. A major part of the railways is already running on electricity, and the transition of road transport is also imminent in the near future.

Key Concepts

• • Three-D path for green energy transition

• • Operation and maintenance of solar PV power plants using digital technologies

• • Role of energy storage in the decarbonization of power grid

• • Microgrid, nanogrid, and vehicle to grid

• • Peer-to-peer energy trading

• • Latest trends in solar cell and module technology

• • Role of smart grid in enabling integration of solar and wind energy sources with the power grid

• • Wind-solar hybrid, floating solar, agro photovoltaics

• • Role of education, training, research and development in successful transition to green energy

Introduction

Rapid transition of the energy system with growing utilization of green and renewable sources has come up with a number of challenges and opportunities. This transition will continue and completely alter the whole energy network. These developments have come up at a time when a number of new and established technologies are available which need to be used and integrated in this changed network. Artificial intelligence (AI), ML, Big Data, cloud computing, blockchain, and so on are some of these important technologies.

Operation and maintenance of solar and wind plants and the role of AI, ML, Big Data and so on; peer-to-peer energy transactions and the role of blockchain in them; grid integration challenges and their solutions; off-grid applications with and without battery storage; handling of PV waste; and solar energy derivatives such as green hydrogen are the areas which are set to play very important roles in the successful transition to the green and distributed energy network.

Apart from these technologies, other important developments are underway, such as solar PV modules of higher efficiency with new technology and material, a new shape, a lesser effect on ambient temperature, requiring less water for cleaning, and so on.