Michael B. Wakin - Research

Low-Dimensional Models for
Signal and Data Processing

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Research Interests

Motivation

Effective techniques for processing and understanding data and information often rely on some sort of model that characterizes the expected behavior of the information. In many cases, the model conveys a notion of constrained structure or conciseness to the information; one may believe, for example, that a length-N data vector (or signal) has few degrees of freedom relative to its size N. Examples include: bandlimited signals, images containing low-dimensional geometric features, or collections of signals observed from multiple viewpoints in a camera or sensor network.

The notion of conciseness is a very powerful assumption, and it suggests the potential for developing highly accurate and efficient data processing algorithms that capture and exploit the inherent structure of the model. In some cases, for example, the signals of interest can be expressed as sparse linear combinations of elements from some dictionary the sparsity of the representation directly reflects the conciseness of the model and permits efficient algorithms for signal processing. Sparsity also forms the core of the emerging theory of Compressed Sensing, which states that a sparse signal can be recovered from a small number of random linear measurements.

In other cases, however, sparse representations may not suffice to truly capture the underlying structure of a signal. Instead, the conciseness of the signal model may impose a low-dimensional geometric (often manifold-like) structure to the signal class as a subset of the high-dimensional ambient signal space. To date, the importance and utility of manifolds for signal processing has been acknowledged largely through a research effort into learning manifold structure from a collection of data points. While these methods have proved effective for certain tasks (such as classification and recognition), they also tend to be quite generic and fail to consider the geometric nuances of specific signal classes.

Projects and Contributions

The purpose of my research is to develop new methods and understanding for signal and data processing based on low-dimensional signal models, with a particular focus on the role of geometry. Below is a list of key topics (with links to selected publications):

Compressed Sensing and Dimensionality Reduction

Manifolds and Multi-Image Processing

Single-Image Processing and Compression

Additional Information & Resources

Email: mwakin (at) mines (dot) e d u