Research

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My research interests are varied, although they generally come under the umbrella of applying mathemical methods to problems in computing. More information can generally be found in my publications. Some general headings are:

Euler Equations on Diffeo Groups

My principal research at the moment is concerned with looking at geometric, analytic, and numerical aspects of generalised Euler equations on diffeomorphism groups. The focus is a better understanding of the equations and their behaviour; this will have beneifts in their utilisation in applications as diverse as medical imaging and fluid dynamics. I am one of three main researchers in the Applied Dynamics Group at Massey, working in this area. (Picture is a diffeomorphic warp of the circle with the green boundary held fixed.)

Machine Learning

My PhD is in machine learning, particularly self-organisation and novelty detection. For more information, including the `Grow When Required' network that was part of the thesis, see here. I still do theoretic work in this area, principally in spline interpolation, but I am part of teams working on applications in pollen recognition and smart homes. You might also be interested in my book.
(Picture from http://www.nada.kth.se/~asa/Game/BigIdeas/ai.html)

Shape Space and Image Analysis

Many of the methods that I am interested in can be applied to images, particularly what is known as 'shape space', the mathematical description of shape (technically, the embeddings of S^1 into R^2 quotiented by the diffeo group of S^1). For example, solutions to the generalised Euler equations on the diffeomorpism group can be applied to image registration, where medical images are brought into alignment to assist in diagnosis of disease. Additionally, splines and other interpolating functions can be used to increase the resolution of images (super-resolution) and to fill in missing information (inpainting). It also enables you to do some fun things. (The picture is an MRI image of my brain; apparently it was in full working order when the scan was taken.)

Complex Systems

I am also interested in complex systems, such as small world and scale-free networks, and particularly in their links to graph theory and computational complexity. This includes work on probabilistic networks. This work is in its early stages.
(Picture from http://paralleldivergence.com/)