Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Room: Exhibit Hall
Luis A. Schneegans III - University of Missouri - St. Louis
Co-Author(s): Victoria Shumakovich, University of Maryland, MA
Have you ever questioned the differential equations behind image processing? Specifically, have you wondered how one could minimize the degradation of an image during image acquisition? Minimizing the Total Variation Flow is the main problem we decided to look at and is modelled by the Total Variation Flow Equation, which was derived from the ρ-Laplacian:∂u/∂t=div(Du/(|Du|)).In this project, we explore radial solutions to the Total Variation Flow (TVF) equation with the help of the Sign Fast Diffusion Equation (SFDE) and prior results in the 1 dimensional case. Specifically for radial solutions, we derive equations and explicit solutions relating to the n-dimensional case. Lastly, we look at how level sets and (time) profiles change.First, we define a solution to the TVF for a radial, continuous, and piecewise smooth function. We then suggest solutions to the TVF for a radially decreasing function that were found in . Second, we look into how the maxima change over time and how future (time) profiles of the equation look. We show two examples with no extrema based on their initial datums. In the last example, we show a different case of how specifically the level sets change over time.The results found looking at explicit radial solutions may suggest that the we could apply this to increasing the saturation in a picture of n≥3 dimensions. Additionally, it may also suggest that the curvatures within a pixel can be fine tuned to include pixels with white and black curvature to create a more fine-grained picture. One last application is that we could apply this to n≥3 dimensions, by inputting more variables such as different colors on a color wheel. Further research topics include finding and using examples that include extrema, as well as working to find a solution to the Total Variation Flow in R^n for an increasing function.References: D.Herrera, M.Korten, N.Saal. New Approximation Results and Barriers for the Total Variation Flow. Preprint, 2016. A. J. Davis, M. K. Korten, A. J. Talbert, and R. C. Tenaglia. Stabilization Times for the Total Variation Flow and the Sign Fast Diffusion Equation. Preprint, 2019. M. Bonforte and A. Figalli. Total variation flow and sign fast diffusion in one dimension. Journal of Differential Equations, 252(8):4455–4480, 2012. M. Korten, C. Moore, P. Salazar. Extinction Times of the Total Variation Flow. Preprint, 2010. D. Herrera, M. Korten, J. Luong, R. McConnell, N. Saal, and J. Vesta. New Approximation Results and Barriers For the Total Variation Flow. Preprint, 2018. F. Andreu-Vaillo, V. Caselles, J. M. Mazón. Parabolic Quasilinear Equations Minimizing Linear Growth Functions Birkhäuser Verlag, 2004.
Funder Acknowledgement(s): We would like to thank Dr. Marianne Korten for being our mentor for this project. We would also like to extend our thanks to Dr. David Yetter and Dr. Kim Klinger-Logan, in addition to Dr. Korten, for running this REU and giving us this opportunity. Lastly, we would like to thank the NSF foundation for funding the SUMaR REU, under the NSF Grant #DMS-1659123.
Faculty Advisor: Marianne Korten, firstname.lastname@example.org
Role: I, along with my partner, researched and expanded the math used in the one-dimensional case to work for R^n. In addition, we wanted to look at the perimeters of level sets in accordance to the radially decreasing function. Our job was to suggest new ways that we could attempt the problem and to learn more about how the maxima change over time in the radially decreasing case. Should the research project continue, we will continue to look at the radially increasing case, which is a more complicated case than the radially decreasing case.