26,840 research outputs found
Directional diffusion models for graph representation learning
In recent years, diffusion models have achieved remarkable success in various
domains of artificial intelligence, such as image synthesis, super-resolution,
and 3D molecule generation. However, the application of diffusion models in
graph learning has received relatively little attention. In this paper, we
address this gap by investigating the use of diffusion models for unsupervised
graph representation learning. We begin by identifying the anisotropic
structures of graphs and a crucial limitation of the vanilla forward diffusion
process in learning anisotropic structures. This process relies on continuously
adding an isotropic Gaussian noise to the data, which may convert the
anisotropic signals to noise too quickly. This rapid conversion hampers the
training of denoising neural networks and impedes the acquisition of
semantically meaningful representations in the reverse process. To address this
challenge, we propose a new class of models called {\it directional diffusion
models}. These models incorporate data-dependent, anisotropic, and directional
noises in the forward diffusion process. To assess the efficacy of our proposed
models, we conduct extensive experiments on 12 publicly available datasets,
focusing on two distinct graph representation learning tasks. The experimental
results demonstrate the superiority of our models over state-of-the-art
baselines, indicating their effectiveness in capturing meaningful graph
representations. Our studies not only provide valuable insights into the
forward process of diffusion models but also highlight the wide-ranging
potential of these models for various graph-related tasks
A Fast Semi-implicit Method for Anisotropic Diffusion
Simple finite differencing of the anisotropic diffusion equation, where
diffusion is only along a given direction, does not ensure that the numerically
calculated heat fluxes are in the correct direction. This can lead to negative
temperatures for the anisotropic thermal diffusion equation. In a previous
paper we proposed a monotonicity-preserving explicit method which uses limiters
(analogous to those used in the solution of hyperbolic equations) to
interpolate the temperature gradients at cell faces. However, being explicit,
this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL)
stability timestep. Here we propose a fast, conservative, directionally-split,
semi-implicit method which is second order accurate in space, is stable for
large timesteps, and is easy to implement in parallel. Although not strictly
monotonicity-preserving, our method gives only small amplitude temperature
oscillations at large temperature gradients, and the oscillations are damped in
time. With numerical experiments we show that our semi-implicit method can
achieve large speed-ups compared to the explicit method, without seriously
violating the monotonicity constraint. This method can also be applied to
isotropic diffusion, both on regular and distorted meshes.Comment: accepted in the Journal of Computational Physics; 13 pages, 7
figures; updated to the accepted versio
Finsler Active Contours
©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TPAMI.2007.70713In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion
In this paper we present our detailed investigations on the nature of the
phase transition in the scalar noise model (SNM) of collective motion. Our
results confirm the original findings of Vicsek et al. [Phys. Rev. Lett. 75
(1995) 1226] that the disorder-order transition in the SNM is a continuous,
second order phase transition for small particle velocities ().
However, for large velocities () we find a strong anisotropy in the
particle diffusion in contrast with the isotropic diffusion for small
velocities. The interplay between the anisotropic diffusion and the periodic
boundary conditions leads to an artificial symmetry breaking of the solutions
(directionally quantized density waves) and a consequent first order transition
like behavior. Thus, it is not possible to draw any conclusion about the
physical behavior in the large particle velocity regime of the SNM.Comment: 13 pages, 11 figure
Directional Bilateral Filters
We propose a bilateral filter with a locally controlled domain kernel for
directional edge-preserving smoothing. Traditional bilateral filters use a
range kernel, which is responsible for edge preservation, and a fixed domain
kernel that performs smoothing. Our intuition is that orientation and
anisotropy of image structures should be incorporated into the domain kernel
while smoothing. For this purpose, we employ an oriented Gaussian domain kernel
locally controlled by a structure tensor. The oriented domain kernel combined
with a range kernel forms the directional bilateral filter. The two kernels
assist each other in effectively suppressing the influence of the outliers
while smoothing. To find the optimal parameters of the directional bilateral
filter, we propose the use of Stein's unbiased risk estimate (SURE). We test
the capabilities of the kernels separately as well as together, first on
synthetic images, and then on real endoscopic images. The directional bilateral
filter has better denoising performance than the Gaussian bilateral filter at
various noise levels in terms of peak signal-to-noise ratio (PSNR)
Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies
The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4×10^5. A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 105 and 1.4×10^5, respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers
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