scholarly journals Asymptotic analysis of the Ginzburg–Landau functional on point clouds

2018 ◽  
Vol 149 (2) ◽  
pp. 387-427 ◽  
Author(s):  
Matthew Thorpe ◽  
Florian Theil
Keyword(s):  
Point Clouds ◽  
Transition Model ◽  
Interaction Potentials ◽  
Total Variation Norm ◽  
Ginzburg Landau ◽  
Data Points ◽  
Variation Norm ◽  
Γ Convergence ◽  
Anisotropic Case ◽  
Classification Type

AbstractThe Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψnwherendenotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

scholarly journals Large data limit for a phase transition model with the p-Laplacian on point clouds

2018 ◽  
Vol 31 (2) ◽  
pp. 185-231 ◽  
Author(s):  
R. CRISTOFERI ◽  
M. THORPE
Keyword(s):  
Phase Transition ◽  
Large Data ◽  
Point Clouds ◽  
Mathematical Tool ◽  
Ginzburg Landau ◽  
Data Points ◽  
Non Local ◽  
Γ Convergence ◽  
Double Well Potential ◽  
Objective Functional

The consistency of a non-local anisotropic Ginzburg–Landau type functional for data classification and clustering is studied. The Ginzburg–Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.

scholarly journals Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2019
Author(s):  
Sameerah Jamal
Keyword(s):  
Fourth Order ◽  
Noise Suppression ◽  
Linear Equations ◽  
Travelling Wave ◽  
Order Partial Differential Equation ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Total Variation Norm ◽  
Wave Solutions ◽  
Variation Norm

In this paper, we discuss travelling wave solutions for image smoothing based on a fourth-order partial differential equation. One of the recurring issues of digital imaging is the amount of noise. One solution to this is to minimise the total variation norm of the image, thus giving rise to non-linear equations. We investigate the variational properties of the Lagrange functionals associated with these minimisation problems.

scholarly journals State Transition for Statistical SLAM Using Planar Features in 3D Point Clouds

Sensors ◽  
2019 ◽  
Vol 19 (7) ◽  
pp. 1614 ◽  
Author(s):  
Amirali Gostar ◽  
Chunyun Fu ◽  
Weiqin Chuah ◽  
Mohammed Hossain ◽  
Ruwan Tennakoon ◽  
...  
Keyword(s):  
Autonomous Vehicle ◽  
Large Body ◽  
Point Clouds ◽  
Substantial Part ◽  
Transition Model ◽  
3D Point Clouds ◽  
Autonomous Cars ◽  
Stochastic Transition ◽  
Vehicle Sensors ◽  
Bayesian Filters

There is a large body of literature on solving the SLAM problem for various autonomous vehicle applications. A substantial part of the solutions is formulated based on using statistical (mainly Bayesian) filters such as Kalman filter and its extended version. In such solutions, the measurements are commonly some point features or detections collected by the sensor(s) on board the autonomous vehicle. With the increasing utilization of scanners with common autonomous cars, and availability of 3D point clouds in real-time and at fast rates, it is now possible to use more sophisticated features extracted from the point clouds for filtering. This paper presents the idea of using planar features with multi-object Bayesian filters for SLAM. With Bayesian filters, the first step is prediction, where the object states are propagated to the next time based on a stochastic transition model. We first present how such a transition model can be developed, and then propose a solution for state prediction. In the simulation studies, using a dataset of measurements acquired from real vehicle sensors, we apply the proposed model to predict the next planar features and vehicle states. The results show reasonable accuracy and efficiency for statistical filtering-based SLAM applications.

scholarly journals Stability with respect to initial conditions in V-norm for nonlinear filters with ergodic observations

2017 ◽  
Vol 54 (1) ◽  
pp. 118-133 ◽  
Author(s):  
Mathieu Gerber ◽  
Nick Whiteley
Keyword(s):  
Initial Conditions ◽  
Stability Result ◽  
Initial Distribution ◽  
Nonlinear Filters ◽  
Initial Condition ◽  
Test Functions ◽  
Total Variation Norm ◽  
Variation Norm ◽  
Prediction Filter ◽  
General Setup

AbstractWe establish conditions for an exponential rate of forgetting of the initial distribution of nonlinear filters in V-norm, allowing for unbounded test functions. The analysis is conducted in an general setup involving nonnegative kernels in a random environment which allows treatment of filters and prediction filters in a single framework. The main result is illustrated on two examples, the first showing that a total variation norm stability result obtained by Douc et al. (2009) can be extended to V-norm without any additional assumptions, the second concerning a situation in which forgetting of the initial condition holds in V-norm for the filters, but the V-norm of each prediction filter is infinite.

scholarly journals AN IDENTIFICATION PROBLEM IN AN URN AND BALL MODEL WITH HEAVY TAILED DISTRIBUTIONS

2009 ◽  
Vol 24 (1) ◽  
pp. 77-97
Author(s):  
Christine Fricker ◽  
Fabrice Guillemin ◽  
Philippe Robert
Keyword(s):  
Basic Problem ◽  
Identification Problem ◽  
Total Variation Norm ◽  
Heavy Tailed Distributions ◽  
Ball Problem ◽  
Original Number ◽  
Explicit Bounds ◽  
Variation Norm ◽  
Heavy Tailed ◽  
Stein Method

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.

scholarly journals Existence and Uniqueness of a Quasistationary Distribution for Markov Processes with Fast Return from Infinity

2014 ◽  
Vol 51 (3) ◽  
pp. 756-768 ◽  
Author(s):  
Servet Martínez ◽  
Jaime San Martín ◽  
Denis Villemonais
Keyword(s):  
Total Variation ◽  
Existence And Uniqueness ◽  
Initial Distribution ◽  
Total Variation Norm ◽  
Birth And Death Processes ◽  
Long Time Behaviour ◽  
Long Time ◽  
Unique Distribution ◽  
Variation Norm ◽  
Quasistationary Distributions

We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N* if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.

scholarly journals Total Variation Applications in Computer Vision

2018 ◽  
pp. 543-570
Author(s):  
Vania Vieira Estrela ◽  
Hermes Aguiar Magalhães ◽  
Osamu Saotome
Keyword(s):  
Computer Vision ◽  
Total Variation ◽  
Total Variation Norm ◽  
Mathematical Background ◽  
Tv Regularization ◽  
Variation Norm ◽  
Image Processing Algorithms ◽  
Processing Algorithms ◽  
Set Up

The objectives of this chapter are: (i) to introduce a concise overview of regularization; (ii) to define and to explain the role of a particular type of regularization called total variation norm (TV-norm) in computer vision tasks; (iii) to set up a brief discussion on the mathematical background of TV methods; and (iv) to establish a relationship between models and a few existing methods to solve problems cast as TV-norm. For the most part, image-processing algorithms blur the edges of the estimated images, however TV regularization preserves the edges with no prior information on the observed and the original images. The regularization scalar parameter λ controls the amount of regularization allowed and it is essential to obtain a high-quality regularized output. A wide-ranging review of several ways to put into practice TV regularization as well as its advantages and limitations are discussed.

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