Definition 1.1 (Binary Operation) Let 𝐺 be a non-empty set. A binary operation on 𝐺 is a function that assigns each ordered pair of elements of 𝐺 an element of 𝐺.

Definition 1.2 (Group) Let 𝐺 be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (𝑎, 𝑏) of elements of 𝐺 an element in 𝐺 denoted by 𝑎𝑏. We say 𝐺 is a group under this operation if the following three properties are satisfied.

DIFFRACTION

Diffraction is a phenomenon in which a light beam bends around the corner of an obstacle and spreads into the geometric shadow of that obstacle.

FRESNEL AND FRAUNHOFER DIFFRACTION

Diffraction can be classified into two categories:

• 1. Fresnel diffraction

• 2. Fraunhofer diffraction

The distinction between these two categories is as follows:

• a. In Fresnel diffraction, the screen and source are at a finite distance from an obstacle. The distances are important in this class. In Fraunhofer diffraction, the source and screen are at an infinite distance from an obstacle. Therefore, inclination is important.

• b. The incident wavefront in Fresnel diffraction is either spherical or cylindrical, whereas the incident wavefront in Fraunhofer diffraction is planar.

• c. In Fresnel diffraction, the central point of the screen is either bright or dark depending on the number of zones, whereas in Fraunhofer diffraction, the central point of the screen is always bright.

FRAUNHOFER DIFFRACTION DUE TO SINGLE SLIT

Let us consider a monochromatic light source of wavelength ƛ placed at the focus of convex lens L1. The collimated rays of plane wavefront are incident on a single-slit AB of width “e.” The un-deviated rays from the slit reaches at point O, and the rays diffracted by an angle θ reach at P on the screen, as shown in Figure 12.1.

In graph theory, planar graphs introduce a fascinating area of study that intersects geometry, topology and network analysis. A graph is called planar if it can be drawn on a plane without any edges crossing each other. This property of planarity has significant implications in fields like circuit design, geographic mapping and urban planning where minimizing crossings leads to more efficient and visually accessible structures. In this chapter we will be covering important topics like the Euler’s formula which provides a basis for understanding the structural constraints of a planar graph. We will also be discussing Kuratowski’s and Wagner’s theorems that provide a criteria for non-planarity. As we will see later in Chapter 9, one of the intriguing mathematical characteristics of planar graphs is that their vertices can be colored by at most four colors.

The subject, Computer Vision, deals with the science ofimparting to a machine or a computer the capabilityof seeing and understanding the environment as wehumans are able to do, and seeks to apply itstheories and models in various applications of ourlife and society. From the late sixties of the lastcentury, there have been efforts in analyzingdigital images captured by a scanner or a camera.Initially, it was the 2-D digital geometry in adiscrete grid of integral coordinate space whichdrew primary attention of the researchers. Inparticular, Prof. Azriel Rosenfeld (1931–2004) ofthe University of Maryland, USA, took a leading andpioneering role in developing theories of digitalpicture processing. Subsequently, the area wasstrengthened by the development and application oftheories of mathematical morphology, textureprocessing, pattern recognition techniques, etc.However, the major development in the theory ofcomputer vision, following the psycho-physiologicalmodels of human vision, happened in the seventies ofthe last century, when Prof. David Marr (1945–1980)of the Massachusetts Institute of Technology (MIT),Cambridge, USA, hypothesized three stages ofprocessing and representation of images by primalsketches consisting of edges, regions, 2.5-Dsketches of the scene, and finally 3-D models.

Over the years, theories of computer vision have beendeveloped from different areas of mathematical andphysical sciences, such as digital geometry,projective geometry, differential geometry, linearand nonlinear systems, human cognition andpsycho-visual perception, color representation andprocessing, computational learning, patternrecognition, etc. As we see, the theoreticalfoundation of the subject has been built fromdifferent domains, and it requires to learn thefundamentals across these disciplines in asystematic and organized manner in the context ofcore agenda of computer vision, which is to solveproblems related to the understanding of a 3-Dscene, static or dynamic, given visual inputs fromimaging systems.

Learning Outcomes

After reading this chapter, the reader will be able to

• Understand the distribution of energy density in a black body radiation as a function of wavelength and temperature

• Derive classical laws of black body radiation such as Wien distribution law and Rayleigh–Jeans law

• Get an idea about the development of quantum theory of radiation

• Understand Planck's quanta postulates and explain the black body radiation spectrum

• Derive Planck's law of black body radiation

• Verify Planck's law of black body radiation experimentally

• Derive Wien distribution law and Rayleigh–Jeans law and explain ultraviolet catastrophe from Planck's law of black body radiation

• Determine the temperature of cosmic microwave background radiation using Planck's law of black body radiation

• Solve numerical problems and multiple choice questions on black body Radiation

11.1 Introduction

Figure 11.1 The whole electromagnetic spectrum. Thermal radiation ranges in frequency from the shortest infrared rays through the visible-light spectrum to the longest ultraviolet rays.

Radiation emitted from the surface of a heated source is known as thermal radiation. In this process, thermal energy is spread out in all directions in the form of electromagnetic radiation and travels directly to its point of absorption at the speed of light. It does not require an intervening medium for its propagation. The wavelength of thermal radiation ranges from the longest infrared rays through the visible-light spectrum to the shortest ultraviolet rays. Such electromagnetic spectrum is shown in Figure 11.1 as a function of frequency. The distribution of radiant energy with their corresponding intensities within various ranges of wavelengths is governed by the temperature of the emitting surface .

In graph theory, independent sets represent collections of vertices that are pairwise nonadjacent, meaning no two vertices within an independent set share an edge. The study of independent sets is often linked to cliques (sets of mutually adjacent vertices) and covering numbers (the smallest set of vertices and edges that cover the entire graph) as they provide contrasting perspectives on how elements within a graph relate to each other.

The world of graph theory is expanding at a pace at which it is hard to keep track of the various disciplines of study that have somehow been irrevocably affected by the techniques owned and created by this versatile subject. However, the beauty of graph theory and its grandeur can also be intimidating for a beginner who wishes to explore its realms. The primary aim of the book is therefore to help a student understand and master the tools and techniques that are inherent to the subject and to serve as a handbook for anyone who wishes to explore the amazing world of graph theory.

We started writing this book with the goal of creating an undergraduate graph theory textbook that would be used by a broad audience, including students and teachers of Mathematics, Computer Science, Economics and perhaps Business Administration. However, mathematical training in proofs and logic is a prerequisite for understanding and following this book.

STATISICAL EXPERIMENTS ENABLE us to make inferences from data about parameters that characterize a population. Generally speaking, inferences may be of two types, namely, deductive inference and inductive inference. Deductive inference pertains to conclusions based on a set of premises (propositions) and their synthesis. Deductive reasoning has a definitive character. For example, all men are mortal (first proposition); Socrates is a man (second proposition); hence, Socrates is mortal (deductive conclusion). On the other hand, inductive inference has a probabilistic character. One conducts an experiment and collects data. Based on this data, certain conclusions are drawn that may have a broader applicability beyond the contours of the particular experiment performed by the researcher. This generalization of the conclusions drawn from the particular experiment constitutes the framework of inductive reasoning. For example, measurement of heights of a small group of people belonging to a certain population is conducted. Based on the calculations of this small sample set, and upon finding that for this small group the average height of men is greater than the average height of women, it is inferred that the men of this population are generally taller than the women.

The formal practice of inductive reasoning dates back to the thesis of Gottfried Wilhelm Leibniz (see Figure 5.1). He was the first to propose that probability is a relation between hypothesis and evidence (data). His thesis was founded on three conceptual pillars: chance (probability), possibilities (realizable random events), and ideas (generalization of inferences by induction). We have encountered the first two concepts in earlier chapters of this textbook. In this chapter, we will delve into the third theme whereby we will discuss methods to draw conclusions from data derived from statistical experiments based on the principles of inductive reasoning.

Biometrics is the scienceand technology of uniquely identifying a person bythe physical, physiological, genomic, or behavioralcharacteristics. For example, the biometric traitsor signatures for unique characterization of aperson may be obtained from fingerprint, palm print,face, iris, retina, shape of ear, voice, signature,gait, vein in the hand, odor, handwriting, DNAsequences, etc. Some of these traits are evidentlyvisible and are often used in our socialinteractions to identify a person. But many of themmay need use of technology and computationalprocessing for extraction of biometriccharacteristic signatures from them and verifyingthem subsequently. For a unique identification of aperson, the biometric data, also referred to as thebiometric signature, should have the properties ofuniqueness and permanence. The uniqueness is thecharacteristics that uniquely identifies anindividual person and permanence implies that itshould remain unchanged throughout the life of theindividual. However, permanence in absolute sense isseldom true in practice. In view of that, it ispragmatic to use those biometric traits, which areexpected to remain mostly unaltered for asignificant period of time. During this period,there may be some marginal deviations that can belargely tolerated for a practical solution.

17.1 | Biometric system

A biometric system isprimarily designed for managing the identity of aperson. Identity management is required in almostevery sphere of social interaction and activitieslike, border control, access control to certainresources, to avail conditional facilities (food,LPG connection, etc.), financial transactions,admission to examinations, certifying thequalification and competence, etc. There may also bevarious related tasks other than identifying aperson like, determining age and gender of anindividual, establishing kinship between twopersons, etc. There are three main generic tasksthat are involved in such a system of identitymanagement.

Classification is a taskof assigning a known category or class to an object.A class is a wellstudied group of objects that is identified by theircommon properties or characteristics. For example,consider the image in Fig. 7.1, where an instance ofa region in the image is denoted by a rectangularbounding box. Here, the task is to classifydifferent regions in the image by consideringvarious patches, as illustrated by a few boundingboxes, to two classes, “human” or “nonhuman”.Likewise a few other examples of imageclassification problem are, detection of pedestriansin an image patch, recognition of a letter given atwo-dimensional (2-D) image pattern, assigning apixel of an image to its foreground or background,finding whether an image captured indoor or outdoor,etc. We may observe here the diversified nature ofclassification problems and by solving them,different types of tasks are performed. Mostly, theclassification problem falls under the supervisedlearning framework, where training samples withappropriate features and class labels are used tolearn a model that is suitable for predictingclasses of the given data. There are variousapproaches for addressing the classification problemlike probabilistic approach, distance basedapproach, discriminant analysis based approach,artificial neural network (ANN) based approach, etc.This chapter introduces four specific techniquesfrom these approaches, namely, Bayesianclassification technique (particularly, naiveBayesian classifier), 𝐾-nearest neighbor (𝐾-NN)classifier, use of linear discriminant functions,and artificial neutral network, respectively.