Half Day Meeting on the 23rd of February 2012

The Mathematics & Information Day is a half-day meeting, centred upon Cambridge, of researchers who are interested in the broad range of modern mathematical methodologies in understanding information-rich phenomena, e.g. data analysis, image and signal processing, medical imaging and compressed sensing.

Organisers:

Anders Hansen, Department of Applied Mathematics & Theoretical Physics (DAMTP)

Arieh Isereles, DAMTP

Carola-Bibiane Schönlieb, DAMTP

Talk abstracts & slides:

Nick Kingsbury (Signal Processing & Communications Group, Department of Engineering)

Complex-valued wavelets, the dual tree, and Hilbert pairs: why these lead to shift invariance and directional multi-dimensional wavelets

Abstract: This talk will discuss some of the interesting mathematical properties of the dual tree wavelet construction, when it is used to generate analytic complex-valued wavelets. We shall show how the requirement for analyticity (zero energy at negative frequencies) implies that the wavelet bases from the two parts of the dual tree should form a Hilbert transform pair, and how this leads to the half-sample delay condition between the lowpass (scaling function) filters of each tree. It will then be shown how this results in approximate shift invariance of the complex wavelet transform, and the elegant Q-shift filter solution to this condition will be given. We will then discuss how the analytic nature of the wavelets in 1-D allows separable wavelet filters in higher dimensional spaces to be highly directionally selective. Finally we will present a wide range of applications for the dual-tree wavelets, including denoising, regularisation, motion / displacement estimation of 2-D and 3-D datasets, and object recognition.

Daniel Holland (Department of Chemical Engineering and Biotechnology)

Sparse sampling in Chemical Engineering, Biochemistry, and Materials Science

Abstract: This talk will present an introduction to applications of sparse sampling in chemical engineering, biochemistry and materials science. I will introduce the basic measurement principle behind each technique and illustrate how the measurements can be modified to exploit ideas from sparse sampling. I will principally focus on compressed sensing, but also present a few examples where Bayesian analysis can lead to even more dramatic reductions in acquisition time. These techniques are now permitting acquisitions an order of magnitude (or more) faster than was previously possible, and therefore enabling us to probe problems that were beyond the range of conventional measurement approaches. I will illustrate the talk using examples including velocity mapping in gas-liquid flows, the structural characterisation of proteins, and nano-metrological characterisation of catalysts.

Peter Nestor (Department of Clinical Neurosciences)

Brain imaging in dementiaswe report what we see but is what we see the truth?

Abstract: Imaging technology has reached a stage of sophistication where it can be applied to address mechanistic questions in dementias. For instance, structure, neural connectivity, metabolism, receptor populations and even pathological deposits can now be imaging non-invasively. This offers the potential to understand both the spatial and temporal relationships of the various elements (e.g. cell loss, toxic deposition of proteins etc.) of these diseases with the potential to therefore tease out causal relationships between pathology and neural degeneration. There are, however, considerable methodological challenges. Different imaging techniques have differing signal-to-noise ratios; furthermore errors in image processing such as tissue segmentation or delineating boundaries can lead to both failures in detection of change and/or identification of spurious relationships. This presentation will highlight some of the problems with particular reference to processing imaging data.

Richard Nickl (Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics)

Statistical inference in high- and infinite-dimensional models

Pushmeet Kohli (Microsoft Research)

Efficient maximum a posterior (MAP) inference for computer vision and beyond

Abstract: Many problems in computer vision and machine learning require inferring the most probable states of certain hidden or unobserved variables. This inference problem can be formulated in terms of minimizing a function of discrete variables. The scale and form of computer vision problems raise many challenges in this optimization task. For instance, functions encountered in vision may involve millions or sometimes even billions of variables. Furthermore, the functions may contain terms that encode very high-order interaction between variables. These properties ensure that the minimization of such functions using conventional algorithms is extremely computationally expensive.

In this talk, I will discuss how many of these challenges can be overcome by exploiting the sparse and heterogeneous nature of discrete optimization problems encountered in real world computer vision problems. Such problem-aware approaches to optimization can lead to substantial improvements in running time and allow us to produce good solutions to many important problems.

Carola-Bibiane Schönlieb (Department of Applied Mathematics and Theoretical Physics)

Mathematical image enhancement in medicine, forensics and the arts

Abstract: In the modern society we encounter digital images in many different situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications. Examples of medical imaging tools are MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography) and CT (computed tomography) for imaging the brain or other organs such as the heart. These imaging tools usually produce noisy or incomplete image data. Hence, before they can be evaluated by doctors, they have to be processed. Keywords in this context are image denoising, image deblurring, image decomposition and image inpainting.

In this talk I will present one of the most successful processing approaches: partial differential equations and variational models. Given a noisy image, its processed (denoised) version is computed as a solution of a PDE or as a minimiser of a functional (variational model). Both of these processes are regularising the given image. In favourable imaging approaches this is done by eliminating high-frequency features (noise) while preserving or even enhancing low-frequency features (object boundaries, edges). After discussing the main principles of this mathematical concept the talk is furnished with computational examples from medical imaging, forensics and the arts.

Acknowledgement:

This workshop is supported by the DAMTP, the Statistics Laboratory (DPMMS), the Cambridge Centre for Analysis (CCA), and Award No. KUK-I1-007- 43, made by King Abdullah University of Science and Technology (KAUST).