45771940
Smoothing Tensor-Valued Images Using Anisotropic Geodesic Diffusion
This paper considers the feature space of DT-MRI as a differential manifold with an affine-invariant metric. We generalise Di Zenzo’s structure tensor to tensor-valued images for edge detection. To improve the quality of the edges, we develop a generalised Perona-Malik method for smoothing tensor images. We demonstrate our algorithm on both synthetic and real DT-MRI data.
21190705
Evolving Spanning Trees Using the Heat Equation
This paper explores how to use the heat kernel to evolve the minimum spanning tree of a graph with time. We use the heat kernel to weight the edges of the graph, and these weights can be computed by exponentiating the Laplacian eigensystem of the graph with time. The resulting spanning trees exhibit an interesting behaviour as time increases. Initially, they are bushy and rooted near the centre of graph, but as time evolves they become string-like and hug the boundary of the graph. We characterise this behaviour using the distribution of terminal nodes with time, and use this distribution for the purposes of graph clustering and image segmentation.
15564869
Heat Kernel Smoothing of Scalar and Vector Image Data
This paper shows how the graph-spectral heat kernel can be used to smooth both gray-scale and color images. We represent images using weighted attributed graphs, and compute the associated Laplacian matrix. Diffusion across this weighted graph-structure with time is captured by the heat-equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigen-system with time. Image smoothing is effected by convolving the heat kernel with the image. The method has the effect of smoothing within regions, but does not blur region boundaries. Experiments and comparisons on standard images illustrate the effectiveness of the method.
14205748
Edge Detection and Anisotropic Diffusion for Tensor-Valued Images
This paper considers feature space of DT-MRI as a differential manifold with an affine-invariant metric, where we generalise Di Zenzo's structure tensor to tensor-valued images for edge detection. To improve the quality of the edges, we smooth out the noise by generalised Perona-Malik method for tensor images. We demonstrate our algorithm on both synthetic and real DT-MRI data.
195348993
A Riemannian Weighted Filter for Edge-sensitive Image Smoothing
1746401
On the Effective Enumerability of NP Problems
In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few best solutions to the instance. Motivated by this, we propose in this paper a new framework to measure the effective enumerability of NP optimization problems. More specifically, given an instance of an NP problem, we consider the parameterized problem of enumerating a given number of best solutions to the instance, and study its average complexity in terms of the number of solutions. Our framework is different from the previously-proposed ones. For example, although it is known that counting the number of k-paths in a graph is #W[1]-complete, we present a fixed-parameter enumeration algorithm for the problem. We show that most algorithmic techniques for fixed-parameter tractable problems, such as search trees, color coding, and bounded treewidth, can be used for parameterized enumerations. In addition, we design elegant and new enumeration techniques and show how to generate small-size structures and enumerate solutions efficiently.
30156477
Examination of load-balancing methods to improve efficiency of a composite materials manufacturing process simulation under uncertainty using distributed computing
Process simulations play an important role in guiding process understanding and development, without requiring costly manufacturing trials. For process design under uncertainty, a large number of simulations is needed for an accurate convergence of the moments of the output distributions, which renders such stochastic analysis computationally intensive. This paper discusses the application of a basic distributed computing approach to reduce the computation time of a composite materials manufacturing process simulation under uncertainty. Specifically, several load-balancing methods are explored and analyzed to determine the best strategies given heterogeneous tasks and heterogeneous networks, especially when the individual task times cannot be predicted.
41201857
A Note on the Number of Perfect Matchings of Bipartite Graphs
Abstract Let G be a bipartite graph with 2n vertices, A its adjacency matrix and K the number of perfect matchings. For plane bipartite graphs each interior face of which is surrounded by a circuit of length 4s + 2, s ϵ {1, 2, …}, an elegant formula, i.e. det A = (−1)nK2, had been rigorously proved by Cvetkovic et al. (1982). For general bipartite graphs, this note contains a necessary and sufficient condition for the above relation to hold. A fast algorithm to check if a plane bipartite graph has such a relation is given.
9458829
Face Tracing Based Geographic Routing in Nonplanar Wireless Networks
Scalable and efficient routing is a main challenge in the deployment of large ad hoc wireless networks. An essential element of practical routing protocols is their accommodation of realistic network topologies. In this paper, we study geographic routing in general large wireless networks. Geographic routing is a celebrated idea that uses the locations of nodes to effectively support routing. However, to guarantee delivery, recent geographic routing algorithms usually resort to perimeter routing, which requires the removal of communication links to get a planar sub-network on which perimeter routing is performed. Localized network planarization requires the wireless network to be a unit-disk graph (UDG) or its close approximation. For networks that significantly deviate from the UDG model, a common case in practice, substantially more expensive and non-localized network planarization methods have to be used. How to make such methods efficiently adaptable to network dynamics, and how to avoid the removal of an excessive number of links that leads to lowered routing performance, are still open problems. To enable efficient geographic routing in general wireless networks, we present face-tracing based routing, a novel approach that routes the message in the faces of the network that are virtually embedded in a topological surface. Such faces are easily recognizable and constructible, and adaptively capture the important geometric features in wireless networks - in particular, holes, -thus leading to very efficient routing. We show by both analysis and si mulations that the face-tracing based routing is a highly scalable routing protocol that generates short routes, incurs low overhead, adapts quickly to network dynamics, and is very robust to variations in network models.
52810037
A Riemannian Weighted Filter for Edge-sensitive Image Smoothing
This paper describes a new method for image smoothing. We view the image features as residing on a differential manifold, and we work with a representation based on the exponential map for this manifold (i.e. the map from the manifold to a plane that preserves geodesic distances). On the exponential map we characterise the features using a Riemannian weighted mean. We show how both gradient descent and Newton's method can be used to find the mean. Based on this weighted mean, we develop an edge-preserving filter that combines Gaussian and median filters of gray-scale images. We demonstrate our algorithm both on direction fields from shape-from-shading and tensor-valued images