1,088 research outputs found
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
On Computing the Average Distance for Some Chordal-Like Graphs
The Wiener index of a graph G is the sum of all its distances. Up to renormalization, it is also the average distance in G. The problem of computing this parameter has different applications in chemistry and networks. We here study when it can be done in truly subquadratic time (in the size n+m of the input) on n-vertex m-edge graphs. Our main result is a complete answer to this question, assuming the Strong Exponential-Time Hypothesis (SETH), for all the hereditary subclasses of chordal graphs. Interestingly, the exact same result also holds for the diameter problem. The case of non-hereditary chordal subclasses happens to be more challenging. For the chordal Helly graphs we propose an intricate O?(m^{3/2})-time algorithm for computing the Wiener index, where m denotes the number of edges. We complete our results with the first known linear-time algorithm for this problem on the dually chordal graphs. The former algorithm also computes the median set
Medians in median graphs and their cube complexes in linear time
The median of a set of vertices of a graph is the set of all vertices
of minimizing the sum of distances from to all vertices of . In
this paper, we present a linear time algorithm to compute medians in median
graphs, improving over the existing quadratic time algorithm. We also present a
linear time algorithm to compute medians in the -cube complexes
associated with median graphs. Median graphs constitute the principal class of
graphs investigated in metric graph theory and have a rich geometric and
combinatorial structure, due to their bijections with CAT(0) cube complexes and
domains of event structures. Our algorithm is based on the majority rule
characterization of medians in median graphs and on a fast computation of
parallelism classes of edges (-classes or hyperplanes) via
Lexicographic Breadth First Search (LexBFS). To prove the correctness of our
algorithm, we show that any LexBFS ordering of the vertices of satisfies
the following fellow traveler property of independent interest: the parents of
any two adjacent vertices of are also adjacent. Using the fast computation
of the -classes, we also compute the Wiener index (total distance) of
in linear time and the distance matrix in optimal quadratic time
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