Partial Difference Operators on Weighted Graphs for Image Processing on Surfaces and Point Clouds

2014 ◽  
Vol 23 (9) ◽  
pp. 3896-3909 ◽  
Author(s):  
Francois Lozes ◽  
Abderrahim Elmoataz ◽  
Olivier Lezoray
Keyword(s):  
Image Processing ◽  
Point Clouds ◽  
Difference Operators ◽  
Weighted Graphs ◽  
Partial Difference

scholarly journals On the game p-Laplacian on weighted graphs with applications in image processing and data clustering

2017 ◽  
Vol 28 (6) ◽  
pp. 922-948 ◽  
Author(s):  
A. ELMOATAZ ◽  
X. DESQUESNES ◽  
M. TOUTAIN
Keyword(s):  
Image Processing ◽  
Laplacian Operator ◽  
Weighted Graphs ◽  
Unified Framework ◽  
Partial Difference ◽  
Interpolation Problems ◽  
Game Theoretic ◽  
Non Local ◽  
Uniqueness And Existence ◽  
Local Average

Game-theoretic p-Laplacian or normalized p-Laplacian operator is a version of classical variational p-Laplacian which was introduced recently in connection with stochastic games called Tug-of-War with noise (Peres et al. 2008, Tug-of-war with noise: A game-theoretic view of the p-laplacian. Duke Mathematical Journal145(1), 91–120). In this paper, we propose an adaptation and generalization of this operator on weighted graphs for 1 ≤ p ≤ ∞. This adaptation leads to a partial difference operator which is a combination between 1-Laplace, infinity-Laplace and 2-Laplace operators on graphs. Then we consider the Dirichlet problem associated to this operator and we prove the uniqueness and existence of the solution. We show that the solution leads to an iterative non-local average operator on graphs. Finally, we propose to use this operator as a unified framework for interpolation problems in signal processing on graphs, such as image processing and machine learning.

scholarly journals Connect the dots: how many random points can a regular curve pass through?

2005 ◽  
Vol 37 (03) ◽  
pp. 571-603 ◽  
Author(s):  
Ery Arias-Castro ◽  
David L. Donoho ◽  
Xiaoming Huo ◽  
Craig A. Tovey
Keyword(s):  
Image Processing ◽  
Point Clouds ◽  
Regular Curve ◽  
Smooth Functions ◽  
Higher Dimensional ◽  
Random Location ◽  
Pass Through ◽  
Lipschitz Graphs

Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as O P (n 1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1] d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in R d may contain as many as O P (n k/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R 2 may pass through at most O P (n 1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

scholarly journals Evaluation of feature-based methods for automated network orientation

2014 ◽  
Vol XL-5 ◽  
pp. 47-54 ◽  
Author(s):  
F.I. Apollonio ◽  
A. Ballabeni ◽  
M. Gaiani ◽  
F. Remondino
Keyword(s):  
Image Processing ◽  
3D Reconstruction ◽  
Processing Time ◽  
Point Clouds ◽  
Large Networks ◽  
Sift Algorithm ◽  
3D Point Clouds ◽  
Network Orientation ◽  
Feature Based ◽  
Automated Image Processing

Every day new tools and algorithms for automated image processing and 3D reconstruction purposes become available, giving the possibility to process large networks of unoriented and markerless images, delivering sparse 3D point clouds at reasonable processing time. In this paper we evaluate some feature-based methods used to automatically extract the tie points necessary for calibration and orientation procedures, in order to better understand their performances for 3D reconstruction purposes. The performed tests – based on the analysis of the SIFT algorithm and its most used variants – processed some datasets and analysed various interesting parameters and outcomes (e.g. number of oriented cameras, average rays per 3D points, average intersection angles per 3D points, theoretical precision of the computed 3D object coordinates, etc.).

On divided difference operators in function algebras

2008 ◽  
Vol 41 (2) ◽  
Author(s):  
Piotr Multarzyński
Keyword(s):  
Divided Difference ◽  
Difference Operators ◽  
Difference Quotient ◽  
Partial Difference ◽  
Function Algebras ◽  
Definition Of ◽  
Product Rules ◽  
Set Of Points ◽  
Tension Structure

AbstractIn this paper we study divided difference operators of any order acting in function algebras. In the definition of difference quotient operators we use a tension structure defined on the set of points on which depend the functions of the algebras considered. In the paper we mention the oportunity for partial difference quotient operators as well as for some purely algebraic definition of divided difference operators in terms of the suitable Leibniz product rules.

PDES Level Set on Weighted Graphs and Applications for Images, Manifold and Point Clouds Processing

2013 ◽  
Vol 46 (11) ◽  
pp. 635-640
Author(s):  
Abdallah EL Chakik ◽  
Xavier Desquesnes ◽  
Abdderahim Elmoataz
Keyword(s):  
Level Set ◽  
Point Clouds ◽  
Weighted Graphs

scholarly journals Determining the Optimal Number of Ground Control Points for Varying Study Sites through Accuracy Evaluation of Unmanned Aerial System-Based 3D Point Clouds and Digital Surface Models

Drones ◽  
2020 ◽  
Vol 4 (3) ◽  
pp. 49 ◽  
Author(s):  
Jae Jin Yu ◽  
Dong Woo Kim ◽  
Eun Jung Lee ◽  
Seung Woo Son
Keyword(s):  
Image Processing ◽  
Point Clouds ◽  
Optimal Number ◽  
Ground Control ◽  
Control Points ◽  
Widespread Application ◽  
Ground Control Points ◽  
3D Point Clouds ◽  
Surface Models ◽  
Study Sites

The rapid development of drone technologies, such as unmanned aerial systems (UASs) and unmanned aerial vehicles (UAVs), has led to the widespread application of three-dimensional (3D) point clouds and digital surface models (DSMs). Due to the number of UAS technology applications across many fields, studies on the verification of the accuracy of image processing results have increased. In previous studies, the optimal number of ground control points (GCPs) was determined for a specific area of a study site by increasing or decreasing the amount of GCPs. However, these studies were mainly conducted in a single study site, and the results were not compared with those from various study sites. In this study, to determine the optimal number of GCPs for modeling multiple areas, the accuracy of 3D point clouds and DSMs were analyzed in three study sites with different areas according to the number of GCPs. The results showed that the optimal number of GCPs was 12 for small and medium sites (7 and 39 ha) and 18 for the large sites (342 ha) based on the overall accuracy. If these results are used for UAV image processing in the future, accurate modeling will be possible with minimal effort in GCPs.

Feature extraction from 3D lidar point clouds using image processing methods

2011 ◽  
Author(s):  
Ling Zhu ◽  
Ashton Shortridge ◽  
David Lusch ◽  
Ruoming Shi
Keyword(s):  
Image Processing ◽  
Feature Extraction ◽  
Point Clouds ◽  
Processing Methods ◽  
3D Lidar
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