arXiv subjects

Bart Smets

Publications and source records attributed to Bart Smets.

4 recordsLinked to original sources

arXiv2022

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at GitHub (https://github.com/NickyvdBerg/DataDrivenTracking).

math.DG↗

arXiv2020

PDE-based Group Equivariant Convolutional Neural Networks

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.

cs.LG↗

arXiv2019

Total Variation and Mean Curvature PDEs on $\mathbb{R}^d \rtimes S^{d-1}$

Total variation regularization and total variation flows (TVF) have been widely applied for image enhancement and denoising. To include a generic preservation of crossing curvilinear structures in TVF we lift images to the homogeneous space $M = \mathbb{R}^d \rtimes S^{d-1}$ of positions and orientations as a Lie group quotient in SE(d). For d = 2 this is called 'total roto-translation variation' by Chambolle & Pock. We extend this to d = 3, by a PDE-approach with a limiting procedure for which we prove convergence. We also include a Mean Curvature Flow (MCF) in our PDE model on M. This was first proposed for d = 2 by Citti et al. and we extend this to d = 3. Furthermore, for d = 2 we take advantage of locally optimal differential frames in invertible orientation scores (OS). We apply our TVF and MCF in the denoising/enhancement of crossing fiber bundles in DW-MRI. In comparison to data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings. We support this by error comparisons on a noisy DW-MRI phantom. We also apply our TVF and MCF in enhancement of crossing elongated structures in 2D images via OS, and compare the results to nonlinear diffusions (CED-OS) via OS.

math.AP↗

arXiv2014

Investigation on Demand Side Management Techniques in the Smart Grid using Game Theory and ICT Concepts

This paper investigates how concepts from game theory and ICT can contribute to solve challenges in demand side management, an important concept in the upcoming smart grid. Demand side management is about modifying the energy load distribution on the demand side, for example in order to reduce peaks in energy usage. This can be done by shifting energy demands where possible. We start with describing a number of smart grid concepts and assumptions (smart meters, pricing, appliance scheduling) and explain the advantages demand side management has. After the introduction of game theoretic concepts, it becomes possible to mathematically describe the demand side management problem. Next step is to solve the mathematical formulation, and show how complex demand side management becomes if the number of energy users increases. By means of distributed ICT algorithms however, it is possible to still find a solution. Based on existing literature, different algorithms are studied. Though results in literature looked promising, several conclusions on convergence of the algorithm in general, and convergence towards the most optimal results in particular are challenged.

cs.DC↗