Mini-Workshop: Analytical and Numerical Methods in Image and Surface Processing

The workshop successfully brought together researchers from mathematical analysis, numerical mathematics, computer graphics and image processing. The focus was on variational methods in image and surface processing such as active contour models, Mumford-Shah type functionals, image and surface denoising based on geometric evolution problems in image and surface fairing, physical modeling of surfaces, the restoration of images and surfaces using higher order variational formulations

Mini-Workshop: Anisotropic Motion Laws

Anisotropic motion laws play a key role in many applications ranging from materials science, biophysics to image processing. All these highly diversified disciplines have made it necessary to develop common mathematical foundations and framworks to deal with anisotropy in geometric motion. The workshop brings together leading experts from various fields to address well-posedness, accuracy, and computational efficiency of the mathematical models and algorithms

Curvature-driven PDE methods for matrix-valued images

Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures in tensor fields, we first generalise Di Zenzo\u27s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts

Statistical exponential formulas for homogeneous diffusion

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula $$u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \,$$ where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by $$\left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by $$\left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly

Nonlinear operators on graphs via stacks

International audienceWe consider a framework for nonlinear operators on functions evaluated on graphs via stacks of level sets. We investigate a family of transformations on functions evaluated on graph which includes adaptive flat and non-flat erosions and dilations in the sense of mathematical morphology. Additionally, the connection to mean motion curvature on graphs is noted. Proposed operators are illustrated in the cases of functions on graphs, textured meshes and graphs of images

Doubly Degenerate Diffuse Interface Models of Anisotropic Surface Diffusion

We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and strong anisotropies and demonstrate the capabilities of the approach for these cases numerically. The proposed model provides a variational and energy dissipative approach for anisotropic surface diffusion, enabling large scale simulations with material-specific parameters.Comment: 15 pages; 6 figure

Lesion boundary segmentation using level set methods

This paper addresses the issue of accurate lesion segmentation in retinal imagery, using level set methods and a novel stopping mechanism - an elementary features scheme. Specifically, the curve propagation is guided by a gradient map built using a combination of histogram equalization and robust statistics. The stopping mechanism uses elementary features gathered as the curve deforms over time, and then using a lesionness measure, defined herein, ’looks back in time’ to find the point at which the curve best fits the real object. We implement the level set using a fast upwind scheme and compare the proposed method against five other segmentation algorithms performed on 50 randomly selected images of exudates with a database of clinician marked-up boundaries as ground truth

Perceptual Color Image Smoothing via a New Region-Based PDE Scheme

In this paper, we present a new color image regularization method using a rotating smoothing filter. This approach combines a pixel classification method, which roughly determines if a pixel belongs to a homogenous region or an edge with an anisotropic perceptual edge detector capable of computing two precise diffusion directions. Using a now classical formulation, image regularization is here treated as a variational model, where successive iterations of associated PDE (Partial Differential Equation) are equivalent to a diffusion process. Our model uses two kinds of diffusion: isotropic and anisotropic diffusion. Anisotropic diffusion is accurately controlled near edges and corners, while isotropic diffusion is applied to smooth regions either homogeneous or corrupted by noise. A comparison of our approach with other regularization methods applied on real images demonstrate that our model is able to efficiently restore images as well as handle diffusion, and at the same time preserve edges and corners well