Graphs and graph polynomials

A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017In this work we study the k-defect polynomials of a graph G. The k defect polynomial is a function in λ that gives the number of improper colourings of a graph using λ colours. The k-defect polynomials generate the bad colouring polynomial which is equivalent to the Tutte polynomial, hence their importance in a more general graph theoretic setting. By setting up a one-to-one correspondence between triangular numbers and complete graphs, we use number theoretical methods to study certain characteristics of the k-defect polynomials of complete graphs. Specifically we are able to generate an expression for any k-defect polynomial of a complete graph, determine integer intervals for k on which the k-defect polynomials for complete graphs are equal to zero and also determine a formula to calculate the minimum number of k-defect polynomials that are equal to zero for any complete graph.XL201

The geometry of colour

This thesis explores the geometric description of animal colour vision. It examines the relationship of colour spaces to behavior and to physiology. I provide a derivation of, and explore the limits of, geometric spaces derived from the notion of risk and uncertainty aversion as well as the geometric objects that enumerate the variety of achievable colours. Using these principles I go on to explore evolutionary questions concerning colourfulness, such as aposematism, mimicry and the idea of aesthetic preference

Monitoring, Modelling and Management of Water Quality

Different types of pressures, such as nutrients, micropollutants, microbes, nanoparticles, microplastics, or antibiotic-resistant genes, endanger the quality of water bodies. Evidence-based pollution control needs to be built on the three basic elements of water governance: Monitoring, modeling, and management. Monitoring sets the empirical basis by providing space- and time-dependent information on substance concentrations and loads, as well as driving boundary conditions for assessing water quality trends, water quality statuses, and providing necessary information for the calibration and validation of models. Modeling needs proper system understanding and helps to derive information for times and locations where no monitoring is done or possible. Possible applications are risk assessments for exceedance of quality standards, assessment of regionalized relevance of sources and pathways of pollution, effectiveness of measures, bundles of measures or policies, and assessment of future developments as scenarios or forecasts. Management relies on this information and translates it in a socioeconomic context into specific plans for implementation. Evaluation of success of management plans again includes well-defined monitoring strategies. This book provides an important overview in this context

A psychophysical investigation of human visual perceptual memory. A study of the retention of colour, spatial frequency and motion visual information by human visual short term memory mechanisms.

The aim of this thesis was to investigate how visual information is organised in perceptual short term memory, with special interest in colour, spatial frequency and velocity. Previous studies of VSTM have indicated the existence of specific memory mechanisms for visual attributes such as orientation, spatial frequency, velocity, contrast and colour. The retention of information in visual short term memory for these basic visual attributes can be disrupted by the presentation of masking stimuli during inter-stimulus intervals (ISIs), which are outside the range of traditional sensory masking. We exploited this memory masking effect in order to examine the organisation of visual information in VSTM. Four groups of experiments were conducted in which participants carried out a delayed discrimination paradigm that employed a two-alternative forced choice (2-AFC) procedure in conjunction with a method of constant stimuli. The fidelity of VSTM was measured by performance markers such as discrimination thresholds and point of subjective equalities. We have found selective memory masking effects, which serve as further evidence in favour of the modular organisation in VSTM, namely, that human visual perceptual memory is based upon multiple, tuned channels in case of colour, spatial frequency and speed, similar to those found in the earliest stages of visual processing for spatial frequency. Moreover, each of these storage mechanisms are tuned to a relatively narrow range of stimulus parameters that are closely linked to visual discrimination mechanisms. These findings add further support to the view that low-level sensory processing mechanisms form the basis for the retention of colour, spatial frequency and velocity information in perceptual memory. We also found evidence for the broad range of transfer of memory masking effects across spatial location, which indicates more long range, long duration interactions between channels that are likely to rely upon contributions from neural processes located in higher visual areas. In conclusion, the experiments presented in this thesis provide significant insight into the organization of visual information in perceptual short term memory.Federation of Ophthalmic and Dispensing Optician

Image processing by region extraction using a clustering approach based on color

This thesis describes an image segmentation technique based on watersheds, a clustering technique which does not use spatial information, but relies on multispectral images. These are captured using a monochrome camera and narrow-band filters; we call this color segmentation, although it does not use color in a physiological sense. A major part of the work is testing the method developed using different color images. Starting with a general discussion of image processing, the different techniques used in image segmentation are reviewed, and the application of mathematical morphology to image processing is discussed. The use of watersheds as a clustering technique in two- dimensional color space is discussed, and system performance illustrated. The method can be improved for industrial applications by using normalized color to eliminate the problem of shadows. These methods are extended to segment the image into regions recursively. Different types of color images including both man made color images, and natural color images have been used to illustrate performance. There is a brief discussion and a simple illustration showing how segmentation can be used in image compression, and of the application of pyramidal data structures in clustering for coarse segmentation. The thesis concludes with an investigation of the methods which can be used to improve these segmentation results. This includes edge extraction, texture extraction, and recursive merging

The role of chromatic texture and 3D shape in colour discrimination, memory colour, and colour constancy of natural objects

The primary goal of this work was to investigate colour perception in a natural environment and to contribute to the understanding of how cues to familiar object identity influence colour appearance. A large number of studies on colour appearance employ 2D uniformly coloured patches, discarding perceptual cues such as binocular disparity, 3D luminance shading, mutual reflection, and glossy highlights are integral part of a natural scene. Moreover, natural objects possess specific cues that help our recognition (shape, surface texture or colour distribution). The aim of the first main experiment presented in this thesis was to understand the effect of shape on (1) memory colour under constant and varying illumination and on (2) colour constancy for uniformly coloured stimuli. The results demonstrated the existence of a range of memory colours associated with a familiar object, the size of which was strongly object-shape-dependent. For all objects, memory retrieval was significantly faster for object-diagnostic shape relative to generic shapes. Based on two successive controls, the author suggests that shape cues to the object identity affect the range of memory colour proportionally to the original object chromatic distribution. The second experiment examined the subject’s accuracy and precision in adjusting a stimulus colour to its typical appearance. Independently on the illuminant, results showed that memory colour accuracy and precision were enhanced by the presence of chromatic textures, diagnostic shapes, or 3D configurations with a strong interaction between diagnosticity and dimensionality of the shape. Hence, more cues to the object identity and more natural stimuli facilitate the observers in accessing their colour information from memory. A direct relationship was demonstrated between chromatic surface representation, object’s physical properties, and identificability and dimensionality of shape on memory colour accuracy, suggesting high-level mechanisms. Chromatic textures facilitated colour constancy. The third and fourth experiments tested the subject’s ability to discriminate between two chromatic stimuli in a simultaneous and successive 2AFC task, respectively. Simultaneous discrimination threshold performances for polychromatic surfaces were only due to low-level mechanism of the stimulus, whereas in the successive discrimination, i.e. when memory is involved, high-level mechanisms were established. The effect of shape was strongly task- dependent and was modulate by the object memory colour. These findings together with the strong interaction between chromatic cues and shape cues to the object identity lead to the conclusion that high level mechanisms linked to object recognition facilitated both tasks. Hence, the current thesis presents new findings on memory colour and colour constancy presented in a natural context and demonstrates the effect of high-level mechanisms in chromatic discrimination as a function of cues to the object identity such as shape and texture. This work contributes to a deeper understanding of colour perception and object recognition in the natural world.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees