PROGRAM | Electrical & Computer Engineering

By: Alejandro Parada-Mayorga Chair: Gonzalo Arce

ABSTRACT

Interesting phenomena in nature can often be captured by graphs since objects and data are invariably inter-related in some sense. Social, financial, ecological networks, and the human brain are a few examples of such networks. Data in these networks reside on irregular or otherwise unordered structures. New data science tools are thus emerging to process signals on graph structures where concepts of algebraic and spectral graph theory are being merged with methods used in computational harmonic analysis. A common problem in these networks is to determine which nodes play the most important role, assuming there is a quantity of interest defined on the network. Graph signal sampling thus becomes essential. In this dissertation, we explore a novel departure from prior work, inspired by sampling patterns in traditional dithering and halftoning. Specifically, we design graph signal sampling techniques that promote the maximization of the distance between sampling nodes on the vertex domain and that produce patterns that are characterized on some subclasses of graphs by a low frequency energy. Sampling patterns with these characteristics are referred to in the spatial dithering literature as blue-noise.

In our previous presentation, the connection between existing theoretical results about sampling signals on graphs and blue noise sampling patterns on graphs was established, showing also how the spectral characteristics of these patterns are shaped by their vertex domain attributes. For the generation of blue noise patterns exploiting their vertex domain characteristics a void and cluster algorithm on graphs was proposed. Numerical experiments show that the reconstruction error obtained with these patterns is similar to the one obtained by the state of the art approaches.

In this talk we state the basis for future research on how to generate blue noise sampling patterns with low complexity algorithms providing numerical results with large graphs where other approaches are not applicable. Additionally, in connection to our results on uniqueness sets for cographs discussed previously, we provide a complete characterization of the uniqueness sets in Threshold graphs. Taking advantage of results for the uniqueness sets of the join between two graphs and the cotree structure of a threshold graph, we determine all the uniqueness sets requiring only the knowledge of the degree of the nodes.

Last, but not least, I will be talking briefly about my research on compressed sensing applications. I will talk about the design of sampling patterns in coded apertures for applications in compressive spectral imaging and compressive X-ray tomosynthesis.

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