Computer vision is a branch of artificial intelligence that involves applications of mathematical methods and computers for machine learning from digital images and videos. Here, we apply computer vision-based methods for optical character recognition (OCR) and image compression. OCR has important applications such as process automation like check clearing, digitizing text and image records for online databases, automated analysis of surveillance camera videos for security, automated reading of text from car license plates in a parking lot, etc. But how can we feed visual information to a computer in a form that it can understand and operate on? To this end, we digitized images into vectorized arrays and analyzed data using vector and scalar projections. Further, we applied algorithms with foundations in linear algebra and wrote programs using Python scientific libraries for optical character recognition and image compression. Using IPython, we characterized color and grayscale images as arrays and implemented singular value decompositions (SVD) and principal component analysis (PCA) for grayscale and color image compression studies and OCR. These studies illustrate how mathematical transformations and data reduction methods can be used for optical character recognition, image compression, identification and encryption. This project elucidates the key role of mathematical modeling for computer vision applications.

Motion algorithms are foundational for effective autonomous robot movement. For surface robotics, one particularly useful algorithm is known as pure pursuit, where a robot follows a point along a path that is a constant distance away from the robot. This work hopes to improve the pure pursuit motion algorithm to account for differences in the robot's features by implementing closed loop full state feedback (FSF) control. In addition, this project aims to provide more abilities to the pure pursuit algorithm, such as specification of angle at each point, allow for moving points, and ensure fast and efficient movements. The algorithm additions are made by modifying the calculations or control loop, and using simulations to verify effectiveness. So far, this work has shown promise by enabling intricate movements while being effective. As a whole, the role of this research is to make pure pursuit more useful and effective for any robot operating on a surface.

The design of quantum computers using Nitrogen vacancies in diamond has renewed interest in providing a microscopic understanding of the properties of the various allotropes of carbon. Here we visualized the crystal structure and electron densities of diamond, graphite, and fullerene to understand the novel structures and bonding in these materials. Using vector calculus-based methods, we computed the bond lengths and bond angles of diamond, graphite, and fullerene. While diamond exhibited sp3 bonding, both graphite and fullerene revealed sp2 bonding. These key changes in structure and bonding gave rise to important differences in their brightness, hardness, electrical conductivity, etc. We computed the adjacency matrix of fullerene and used that to understand the network connectivity. These studies provided an atomic level understanding of the structure, bonding, adjacency matrix, and network connectivity in these materials which form essential inputs and aid in the design of quantum computers.