As the demand for a low-carbon and environmentally friendly future increases, so does the importance of mineral and metal commodities. The production of solar panels, wind turbines, energy storage systems and other green technologies require large quantities of minerals and rare earth metals. Natural Resources Canada noted that in 2019, Canada was a global leading producer in minerals required for green technology including graphite, nickel, cobalt, and others [1]. While mineral production continues to rise year over year, the ore grade, i.e., the concentration of a desired material, of multiple common minerals continues to decline. To liberate valuable minerals from low ore grade deposits size reduction processes such as crushing and grinding are required; however, these processes account for over half of all energy consumption on the average mine. As mines are typically remote, fossil fuels are normally used as the main energy source, producing large amounts greenhouse gases, necessitating the need for more efficient size reduction processes. This could be accomplished by predicting how a particular orebody would break. With the surge in image sensing and computing technologies at mining sites many researchers are exploring ore texture and processability characteristics of the ore body. If distinct processability characteristics change based on ore textural feature from a 2D image, then general trends for optimal size reduction of orebodies of similar texture can developed. This work builds on previous work by simulating ore breakage through the superimposition of a predetermined fragmentation pattern, called a mask, onto multiple ore textures. Synthetic, periodic black and white 2D ore textures were created to find a link between simple textural features such as different mineral grain shape, size, and orientation and processability characteristics. A Monte Carlo simulation was performed to generate a large quantity of realistic product particles using the Voronoi tessellations masking technique. To assess the processability of different textures, the percentage area distribution of valuable minerals of each ore texture was compared across the complete range of particle sizes. The valuable mineral percentage area distributions were analyzed for rate and shape of the distribution as particle size decreases, with noticeable differences between textures. The distributions were also parameterized using a two-beta mixture distribution model, expanding on the traditional one beta model developed by King [2,3,4]. These distributions can eventually help the mining industry make informed decisions on how much grinding and crushing will be required to liberate desired minerals from waste rock.

In this thesis, we develop a new computational method using Voronoi vertex model and energy to describe the cell-cell interaction among the epithelial tissue. Several studies and simulations will be showed such as equilibrium states, wound closure process, and abnormal growth. We also perform analysis on circular epithelial wound closure process.

In this thesis, we develop a new computational method using Voronoi vertex model and energy to describe the cell-cell interaction among the epithelial tissue. Several studies and simulations will be showed such as equilibrium states, wound closure process, and abnormal growth. We also perform analysis on circular epithelial wound closure process.

By combining the high-order accuracy of spectral expansions with the locality and geometric flexibility of finite elements, spectral elements are an attractive option for the next generation of numerical climate models. Crucial to their construction is the configuration of nodes in an element — casual placement leads to polynomial fits exhibiting Runge phenomena manifested by wild spatial oscillations. I provide highorder triangular elements suitable for incorporation into existing spectral element codes. Constructed from a variety of measures of optimality, these nodes possess the best interpolation error norms discovered to date. Motivated by the need to accurately determine these error norms, I present an optimization method suitable for finding extrema in a triangle. It marries a branch and bound algorithm to a quadtree smoothing scheme. The resulting scheme is both robust and efficient, promising general applicability. In order to qualitatively evaluate these nodal distributions, I introduce the concept of a Lagrangian Voronoi tessellation. This partitioning of the triangle illustrates the regions over which each node dominates. I argue that distant and disconnected regions are undesirable as they exhibit a non-physical influence. Finally, I have discovered a link between point distributions in the simplex and on the hypersphere. Through a simple transformation, a distance metric is defined permitting the construction of Voronoi diagrams and the calculation of mesh norms.

Coordinated missions with platoons of autonomous agents are rapidly becoming popular because of technological advances in computing, networking, miniaturization and combination of electromechanical systems. These multi-agents networks coordinate their actions to perform challenging spatially-distributed tasks such as search, survey, exploration, and mapping. Environmental monitoring and locational optimization are among the main applications of the emerging technology of wireless sensor networks where the optimality refers to the assignment of sub-regions to each agent, in such a way that a suitable coverage metric is maximized. Usually the coverage metric encodes a distribution of risk defined on the area, and a measure of the performance of individual robots with respect to points inside the region of interest. The risk density can be used to quantify spatial distributions of risk in the domain. The solution of the optimal control problem in which the risk measure is not time varying is well known in the literature, with the optimal con figuration of the robots given by the centroids of the Voronoi regions forming a centroidal Voronoi tessellation of the area. In other words, when the set of mobile robots converge to the corresponding centroids of the Voronoi tessellation dictated by the coverage metric, the coverage itself is maximized. In this work, it is considered a time-varying risk density evolving according to a diffusion equation with varying boundary conditions that quantify a time-varying risk on the border of the workspace. Boundary conditions model a time varying flux of external threats coming into the area, averaged over the boundary length, so that rather than considering individual kinematics of incoming threats it is considered an averaged, distributed effect. This approach is similar to the one commonly adopted in continuum physics, in which kinematic descriptors are averaged over spatial domain and suitable continuum fields are introduced to describe their evolution. By adopting a first gradient constitutive relation between the flux and the density, a simple diffusion equation is obtained. Asymptotic convergence and optimality of the non-autonomous system are studied by means of Barbalat's lemma and connections with varying boundary conditions are established. Some criteria on time-varying boundary conditions and evolution are established to guarantee the stabilities of agents' trajectories. A set of numerical simulations illustrate theoretical results.

This work deals with the description and analysis of image data, which related to the initial stages of the thin film growth. The introductory retrieval section includes a description of thin films and methods of their deposition. The following part is an overview of the growth models of thin layers. The heart of my thesis is the analysis and modification of morphological methods and interpretation of their results. The emphasis is placed on the statistical aspect of methods and their optimal implementation due to the accuracy of the results. The work shows how to modify the radial distribution function and methods based on so-called Voronoi and Delaunay triangulation tessellation so that they can better affect the character of test data. New methods are tested both on the experimental and model data. Then we examine their robustness, sensitivity and their mutual independence. At the conclusion it is introduced and analyzed a new model of thin film growth.

Voronoi tessellation can be used within a couple of different fields. Some of these fields include healthcare, construction and urban planning. Since Voronoi tessellations are used in multiple fields, it is motivated to know the strengths and weaknesses of the algorithms used to generate them, in terms of their efficiency. The objectives of this thesis are to compare two CPU algorithm implementations for Voronoi tessellation in regards to execution time and see which of the two is the most efficient. The algorithms compared are The Bowyer-Watson algorithm and Fortunes algorithm. The Fortunes algorithm used in the research is based upon a pre-existing Fortunes implementation while the Bowyer-Watson implementation was specifically made for this research. Their differences in efficiency were determined by measuring their execution times and comparing them. This was done in an iterative manner, where for each iteration, the amount of data to be computed was continuously increased. The results show that Fortunes algorithm is more efficient on the CPU without using any acceleration techniques for any of the algorithms. It took 70 milliseconds for the Bowyer-Watson method to calculate 3000 input points while Fortunes method took 12 milliseconds under the same conditions. As a conclusion, Fortunes algorithm was more efficient due to the Bowyer-Watson algorithm doing unnecessary calculations. These calculations include checking all the triangles for every new point added. A suggestion for improving the speed of this algorithm would be to use a nearest neighbour search technique when searching through triangles.

The methods presented in this study are aimed at the identification of subpopulations (clones) of genetically similar cells within tissue samples through measurement of loci-specific Fluorescence in-situ hybridization (FISH) spot signals for each nucleus and analyzing cell spatial distributions by way of Voronoi tessellation and Delaunay triangulation to robustly define cell neighbourhoods. The motivation for the system is to examine lung cancer patient for subpopulations of Non-Small Cell Lung Cancer (NSCLC) cells with biologically meaningful gene copy-number profiles: patterns of genetic alterations statistically associated with resistance to cis-platinum/vinorelbine doublet chemotherapy treatment. Current technologies for gene-copy number profiling rely on large amount of cellular material, which is not always available and suffers from limited sensitivity to only the most dominant clone in often heterogeneous samples. Thus, through the use of FISH, the detection of gene copy-numbers is possible in unprocessed tissues, allowing identification of specific tumour clones with biologically relevant patterns of genetic aberrations. The tissue-wide characterization of multiplexed loci-specific FISH signals, described herein, is achieved through a fully automated, multicolour fluorescence imaging microscope and object segmentation algorithms to identify cell nuclei and FISH spots within. Related tumour clones are identified through analysis of robustly defined cell neighbourhoods and cell-to-cell connections for regions of cells with homogenous and highly interconnected FISH spot signal characteristics. This study presents experiments which demonstrate the system’s ability to accurately quantify FISH spot signals in various tumour tissues and in up to 5 colours simultaneously or more through multiple rounds of FISH staining. Furthermore, the system’s FISH-based cell classification performance is evaluated at a sensitivity of 84% and specificity 81% and clonal identification algorithm results are determined to be comparable to clone delineation by a human-observer. Additionally, guidelines and procedures to perform anticipated, routine analysis experiments are established.

The methods presented in this study are aimed at the identification of subpopulations (clones) of genetically similar cells within tissue samples through measurement of loci-specific Fluorescence in-situ hybridization (FISH) spot signals for each nucleus and analyzing cell spatial distributions by way of Voronoi tessellation and Delaunay triangulation to robustly define cell neighbourhoods. The motivation for the system is to examine lung cancer patient for subpopulations of Non-Small Cell Lung Cancer (NSCLC) cells with biologically meaningful gene copy-number profiles: patterns of genetic alterations statistically associated with resistance to cis-platinum/vinorelbine doublet chemotherapy treatment. Current technologies for gene-copy number profiling rely on large amount of cellular material, which is not always available and suffers from limited sensitivity to only the most dominant clone in often heterogeneous samples. Thus, through the use of FISH, the detection of gene copy-numbers is possible in unprocessed tissues, allowing identification of specific tumour clones with biologically relevant patterns of genetic aberrations. The tissue-wide characterization of multiplexed loci-specific FISH signals, described herein, is achieved through a fully automated, multicolour fluorescence imaging microscope and object segmentation algorithms to identify cell nuclei and FISH spots within. Related tumour clones are identified through analysis of robustly defined cell neighbourhoods and cell-to-cell connections for regions of cells with homogenous and highly interconnected FISH spot signal characteristics. This study presents experiments which demonstrate the system’s ability to accurately quantify FISH spot signals in various tumour tissues and in up to 5 colours simultaneously or more through multiple rounds of FISH staining. Furthermore, the system’s FISH-based cell classification performance is evaluated at a sensitivity of 84% and specificity 81% and clonal identification algorithm results are determined to be comparable to clone delineation by a human-observer. Additionally, guidelines and procedures to perform anticipated, routine analysis experiments are established.

The motivation for this work comes from physics, when dealing with microstructures of polycrystalline materials. An adequate probabilistic model is a three-dimensional (3D) random tessellation. The original contribution of the author is dealing with the Gibbs-Voronoi and Gibbs- Laguerre tessellations in 3D, where the latter model is completely new. The energy function of the underlying Gibbs point process reflects interactions between geometrical characteristics of grains. The aim is the simulation, parameter estimation and degree-of-fit testing. Mathematical background for the methods is described and numerical results based on simulated data are presented in the form of tables and graphs. The interpretation of results confirms that the Gibbs-Laguerre model is promising for further investigation and applications.