VIGRE Presentations, Thursday 17 April 2003
|
Image DeNoising using Nonlinear Total Variation and Anisotropic Heat Diffusion
Research with Professor Curry
Noise removal is essential in fields which rely on information transfer via images. Unfortunately most of these images are inherently tainted by some amount of noise. The most common types of noise are salt & pepper, speckle, Gaussian, and Poisson noise. Many noise removing methods exist, each having their own individual characteristics. Some algorithms are specifically defined for certain types of noise while others are broader in scope. We will present the very powerful denoising methods of Nonlinear Total Variation and Anisotropic Heat Diffusion. Both algorithms are very powerful, and each algorithm has traits which make it preferable over the other depending on the denoising task. We will discuss the concepts, mathematical theory, implementation, and results from both denoising algorithms that we have examined.
Research in Computational Mathematics Exploring and Implementing Finite Element Methods
Research with Professor Manteuffel and Professor McCormick
The presentation will encompass the process of learning and implementing finite element methods, beginning with brief descriptions of Galerkin and First Order Systems Least Squares (FOSLS) methods. The presentation will then detail a high level decomposition of the implementation procedures and describe complexities associated with these two methods. Finally, enhancements to an already implemented partial differential equation (PDE) solver will be discussed. The PDE solver (FOSPACK)* uses FOSLS method and is heavily used by the research group members. In order to make the solver extensible to a larger number of problems, the accuracy and generality needs to be improved. This means modifications to the program. A focus of the presentation is one of these modifications. It includes a module that formulates element node basis functions, derivatives to these basis functions and Gaussian quadrature node locations for varied element spaces and integration accuracy.
Mathematical Model of Dispersion Using Lyapunov Exponents
Research with Professor Julien
Understanding the dynamics of systems involving dispersion of passive scalars is important in many fields of engineering and science. Dispersion can include the transport of particles such as contaminants or represent a property of the system such as heat. Ocean dynamics, weather patterns, engines, reactors, and rocket boosters all undergo mixing in different forms, and understanding the dynamics behind these systems is important for prediction of the behavior of the system. Mathematical models provide valuable insight into the complex transport processes that can occur in physical and engineering systems. Good models allow scientists and engineers to more accurately predict ocean currents and weather patterns as well as allowing them to build more efficient reactors and engines. With Dr. Keith Julien and Paul Mullowney, I am studying ways to numerically analyze dispersion in dynamical systems using Lyapunov exponents. I will present our findings of the application of this method to a new class of analytic three-dimensional rotating fluids.
VIGRE Presentations, Thursday 24 April 2003
|
Image Segmentation using Active Contours
Research with Professor Curry
A common problem in image processing is to break an image into its constituent
parts. This can be done based on various criteria including texture, color,
and gradient information. In this presentation we will discuss and demonstrate
a segmentation algorithm proposed by Chan and Vese based on the Mumford-Shah
segmentation problem. The algorithm is formulated as a level set method.
We will demonstrate how the benefits of the level set method, including automatic
changes in topology, are applicable in the segmentation of several test images.
We then will demonstrate the behaviour on two classes of real-world images,
faces and images generated from an electron microscope.
Research with Professor Curry and Professor Norris
There are many different steps involved in the process of making bread
at the Great Harvest Bread Company. One of the first steps is to mill the
wheat, which produces flour. The grinding of the stones used in the mill
adds heat to the flour. Before the flour can be used, it must be returned
to room temperature. Therefore, Scott the Baker places the flour into large
buckets with five copper tubes, which help speed up the cooling process.
Mathematically, the cooling of flour over time can be described using partial
differential equations. We will discuss how Fourier series and heat transfer
ideas and methods can be used to model and analyze the temperature of the
flour. We will also discuss the mathematical and physical significance
of the copper tubes and what other steps could be taken to further speed
up the cooling.
Computational Aspects of PDE's
Research with Luke Olson
For this VIGRE research project, we study computational aspects of the numerical solution to partial differential equations. Numerical approximation of PDE's is an enormous area of active research with applications in many fields. In particular, we are interested in investigating time dependent PDE's of hyperbolic type. We have initiated a study of the stability of numerical methods for first order hyperbolic PDEs and the computational complexities involved in arriving at the time dependent solutions. We have also identified several projects of for future research.