97 research outputs found
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
Diffusion maps tailored to arbitrary non-degenerate Ito processes
We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling
Perturbation of the Eigenvectors of the Graph Laplacian: Application to Image Denoising
The original contributions of this paper are twofold: a new understanding of
the influence of noise on the eigenvectors of the graph Laplacian of a set of
image patches, and an algorithm to estimate a denoised set of patches from a
noisy image. The algorithm relies on the following two observations: (1) the
low-index eigenvectors of the diffusion, or graph Laplacian, operators are very
robust to random perturbations of the weights and random changes in the
connections of the patch-graph; and (2) patches extracted from smooth regions
of the image are organized along smooth low-dimensional structures in the
patch-set, and therefore can be reconstructed with few eigenvectors.
Experiments demonstrate that our denoising algorithm outperforms the denoising
gold-standards
Level set and PDE methods for visualization
Notes from IEEE Visualization 2005 Course #6, Minneapolis, MN, October 25, 2005. Retrieved 3/16/2006 from http://www.cs.drexel.edu/~david/Papers/Viz05_Course6_Notes.pdf.Level set methods, an important class of partial differential equation
(PDE) methods, define dynamic surfaces implicitly as the level set (isosurface)
of a sampled, evolving nD function. This course is targeted for
researchers interested in learning about level set and other PDE-based
methods, and their application to visualization. The course material will
be presented by several of the recognized experts in the field, and will
include introductory concepts, practical considerations and extensive
details on a variety of level set/PDE applications.
The course will begin with preparatory material that introduces the
concept of using partial differential equations to solve problems in
visualization. This will include the structure and behavior of several
different types of differential equations, e.g. the level set, heat and
reaction-diffusion equations, as well as a general approach to developing
PDE-based applications. The second stage of the course will describe the
numerical methods and algorithms needed to implement the mathematics
and methods presented in the first stage, including information on
implementing the algorithms on GPUs. Throughout the course the
technical material will be tied to applications, e.g. image processing,
geometric modeling, dataset segmentation, model processing, surface
reconstruction, anisotropic geometric diffusion, flow field post-processing
and vector visualization.
Prerequisites:
Knowledge of calculus, linear algebra, computer graphics, visualization,
geometric modeling and computer vision. Some familiarity with
differential geometry, differential equations, numerical computing and
image processing is strongly recommended, but not required
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