A mathematical model of the early visual system and border completion via subriemannian Hamiltonian geodesics

In this thesis, we aim to discuss a simple mathematical model for the edge detection mechanism and the boundary completion problem in the human brain in a differential geometry framework. We describe the columnar structure of the primary visual cortex as the fiber bundle R2 × S1, the orientation bundle, and by introducing a first vector field on it, explain the edge detection process. Edges are detected through a lift from the domain in R2 into the manifold R2 × S1 and are horizontal to a completely non-integrable distribution. Therefore, we can construct a subriemannian structure on the manifold R2 × S1, through which we retrieve perceived smooth contours as subriemannian geodesics, solutions to Hamilton’s equations. To do so, in the first chapter, we illustrate the functioning of the most fundamental structures of the early visual system in the brain, from the retina to the primary visual cortex. We proceed with introducing the necessary concepts of differential and subriemannian geometry in chapters two and three. We finally implement our model in chapter four, where we conclude, comparing our results with the experimental findings of Heyes, Fields, and Hess on the existence of an association field.

Mean curvature flow in SE (2) and applications to visual perception

Our goal in this thesis is to provide a result of existence of the degenerate non-linear, non-divergence PDE which describes the mean curvature flow in the Lie group SE(2) equipped with a sub-Riemannian metric. The research is motivated by problems of visual completion and models of the visual cortex.

Correlation between urban form and PM2.5 concentration — Evidence from the New York metropolitan Area and Shanghai city

With the development of the economy and society, air pollution has posed a huge threat to public health around the world, especially to people who live in urban areas. Typically, urban development patterns can be roughly divided into compact cities and urban sprawl. In recent years, the relationship between urban form and air quality (especially PM2.5) is gaining more and more attention from urban planners, environmentalists, and governments. This study is focusing on The New York metropolitan area and Shanghai city, which are both megacities but with different urban spatial forms. For both study areas,there are five main variables to measure the urban form metrics, naming Population Density, Artificial Land Area Per Ten Thousand People, Road Density, Green Land Area Ratio and Artificial Land Area Ratio. In addition, considering the impact of economic activities and public transportation, GDP per capita, Number of bus stop and Number of subway station are used as control variables. Based on the results of regression, a megacity like the New York metropolitan area with urban sprawl shows a low spatial correlation on PM2.5 concentration. Meanwhile, almost all the spatial form indicators effect on PM2.5 concentration is not significant. However, a compact megacity like Shanghai shows a diametrically opposite result. Urban form, especially population density, has a strong relationship with PM2.5 concentration. It can be predicted that a reduction in population density would lead to significant improvements on decrease the PM2.5 concentration in Shanghai. Meanwhile, increasing the ratio of green land and construction area per capita will get a positive influence on reducing PM2.5 concentration as well. Road density is not a significant factor for a megacity in both two urban forms. The way and type of energy used by vehicles on megacities maybe more critical.