Signal Processing - March 2016 - 103

(a)

et al. [26], and a recent book with more engineering flavor
by Robinson [37].
This is but a snapshot of the flexible potential of topological analysis, and the afforded robustness together with an
amenability to analyze large data sets constitute remarkable
capabilities of a topological paradigm in signal analysis, and
much awaits to be explored by the signal processing research
community, particularly when distributed algorithms, as demonstrated in some of the examples herein, are needed.

(b)

figure 11. Reducing coauthorship networks using topological methods:
(a) the coauthorship network CNCTA and (b) the reduced network CNCTA.

(a)

(b)

figure 12. A simple homotopy collapse. In (a), the diagonal edges collapse into the paired 2-simplex, resulting in the complex shown in (b).
Note that a homotopy collapse preserves topological features.

here may be unstable to disturbances in the data, i.e., it is
possible that the generated homological spaces experience a
nontrivial change in rank when the data are subjected to an
arbitrarily small disturbance in the input data. While the applications chosen here are impervious to this effect, this will not
necessarily be true for all applications. Persistent homology
offers a stable topological summary of noisy data [7].
The "Simplicial Homology" section presents a view of
rank of homology spaces as a generalization of the number of
clusters in the data to higher-order features such as nonvanishing holes, voids, etc. Likewise, persistent homology may
be viewed as a higher-order analog of hierarchical clustering,
which is stable and known to have more interesting properties
than hard clustering [27] and has led, in the past few years to
many interesting applications in engineering [17]. For example, an excellent application of topology in clinical medicine
may be found in [35].
As topological methods provide tools for analysis of
various data shapes, there are several application avenues in
computer vision and image processing. Some of the examples include graphical representation of gray-scale images
[39], deformation-invariant models for digital images [20],
[31], shape segmentation [41], and motion analysis [46]. As
a further illustration of the scope of this topic, we refer the
readers to applications such as comparison of maps [1],
graph comparison [11], localization [38], text mining [47],
and distributed trees for high-performance computing [33].
Various books have been recently published in this area of
research, including a great introduction by Edelsbrunner
and Harer [16], a concise book by Zomorodian [50], a more
specific book about computational homology by Kaczynski

Acknowledgments
We profusely thank Vin De Silva first for his inspirational and
pioneering work in topological data analysis and also for his
encouragement and advice throughout this project, with intuitive graphical examples at times. We would also like to thank
the anonymous referees for their excellent contributions that
contributed significantly to the final version of this article.
This article is the result of research work that was generously supported by DTRA under HDTRA1-08-1-0024.

Authors
Hamid Krim (ahk@ncsu.edu) received his B.S. and M.S.
degrees in electrical engineering from Washington University
and his Ph.D. degree in electrical and computer engineering
from Northeastern University. Following his tenure as a
member of technical staff at AT&T Bell Labs, he has worked
in the area of telephony and digital communication systems/
subsystems. He became a U.S. National Science Foundation
(NSF) postdoctoral scholar at Foreign Centers of Excellence
(Laboratoire des Signaux et Systèmes Supelec/University of
Orsay, Paris, France) in 1991. Subsequently, he was a research
scientist at the Laboratory for Information and Decision
Systems, Massachusetts Institute of Technology, Cambridge,
performing/supervising research in his area of interest, and he
later joined the faculty of the Electrical and Computer
Engineering Department at North Carolina State University in
Raleigh. He is an original contributor and now an affiliate of
the Center for Imaging Science sponsored by the U.S. Army.
He also is a recipient of the NSF Career Young Investigator
Award. He was on the editorial board of IEEE Transactions on
Signal Processing, IEEE Transactions on Signal and
Information Processing over Networks, and IEEE Signal
Processing Magazine, and regularly contributes to the IEEE
Signal Processing Society in various ways. He is a member of
the Society for Industrial and Applied Mathematics and Sigma
Xi. His research interests include statistical signal processing
and mathematical modeling, with a keen emphasis on applications. He is a Fellow of the IEEE.
Thanos Gentimis (gentimisth@gmail.com) received
his bachelor's and master's degrees in theoretical mathematics from the National and Kapodistrian University of
Athens, Greece, in 2002 and 2005, respectively. In 2011,
he received his Ph.D. degree in algebraic topology at the
University of Florida. He was a postdoctoral researcher at
the Electrical and Computer Engineering Department of
North Carolina State University, Raleigh. He is an assistant

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March 2016

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