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Filtering on Graphs

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Discrete Calculus

Abstract

Measured data often includes noise. A data point measured in isolation offers little opportunity to tease signal apart from noise. However, this separation of noise from the signal becomes more possible when multiple data points are acquired which have a relationship with each other. A spatial relationship, such as the edge set of a graph, permits the use of the collective data acquisition to make better decisions about the true data underlying each measurement. This process whereby the spatial relationships of the data are used to provide better estimates of the noiseless data is called a filtering or a denoising process. In this chapter, we outline the assumptions used to justify spatial filtering, describe the equivalent of Fourier analysis on a general graph and discuss how different parameter settings of a small number of variational approaches to filtering lead to a large number of commonly used filters. Although our focus in this chapter is on the filtering of node data (0-cochains), we also discuss how these techniques may be applied to the filtering of edge data (i.e., flows, or 1-cochains) and to the filtering of data associated with higher-dimensional cells.

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Notes

  1. 1.

    Note that, while circulant matrices represent circular convolution, Toeplitz matrices, which comprise a distinct class of matrices, represent linear convolution. A thorough treatment of these matrices is available in [172].

  2. 2.

    Note that ‘λ’ is often used to represent an eigenvalue (e.g., Lemma 5.1). We follow Taubin’s notation for his λ–μ algorithm by using ‘λ’ as a parameter when discussing Taubin’s algorithm.

  3. 3.

    Note that in image processing the term edge is used to mean discontinuity (e.g., “edge detection”). However, since the context of this entire book is the analysis/processing of graphs (complexes) and data defined on graphs, we reserve the word edge to refer strictly to a 1-cell (i.e., we use edge in the sense of graph theory).

  4. 4.

    The term energy is used throughout the book to represent an objective function which is optimized to produce a useful application-specific solution. In this case, the solution represents the filtered (denoised) signal. Although the term energy is not generally intended to have a physical relationship to energy, note that the energy described in (5.10b) is actually the power dissipation for an electric circuit (when \(\boldsymbol{\mathsf {x}}\) represents the electrical potentials at every node), as given in Chap. 3.

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Authors and Affiliations

  1. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540-6632, USA

    Dr. Leo J. Grady

  2. Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA

    Dr. Jonathan R. Polimeni

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  1. Dr. Leo J. Grady
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  2. Dr. Jonathan R. Polimeni
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Correspondence to Leo J. Grady .

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Grady, L.J., Polimeni, J.R. (2010). Filtering on Graphs. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_5

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  • DOI: https://doi.org/10.1007/978-1-84996-290-2_5

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84996-289-6

  • Online ISBN: 978-1-84996-290-2

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