A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Disparity and Optical Flow Partitioning Using Extended Potts Priors

This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notation of asymptotically level stable functions we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of minimizers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method

Disparity map generation based on trapezoidal camera architecture for multiview video

Visual content acquisition is a strategic functional block of any visual system. Despite its wide possibilities, the arrangement of cameras for the acquisition of good quality visual content for use in multi-view video remains a huge challenge. This paper presents the mathematical description of trapezoidal camera architecture and relationships which facilitate the determination of camera position for visual content acquisition in multi-view video, and depth map generation. The strong point of Trapezoidal Camera Architecture is that it allows for adaptive camera topology by which points within the scene, especially the occluded ones can be optically and geometrically viewed from several different viewpoints either on the edge of the trapezoid or inside it. The concept of maximum independent set, trapezoid characteristics, and the fact that the positions of cameras (with the exception of few) differ in their vertical coordinate description could very well be used to address the issue of occlusion which continues to be a major problem in computer vision with regards to the generation of depth map

Joint Image Reconstruction and Segmentation Using the Potts Model

We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from $7$ angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data