Watershed of a Continuous Function

Special issue on Mathematical Morphology.International audienceThe notion of watershed, used in morphological segmentation, has only a digital definition. In this paper, we propose to extend this definition to the continuous plane. Using this continuous definition, we present the watershed differences with classical edge detectors. We then exhibit a metric in the plane for which the watershed is a skeleton by influence zones and show the lower semicontinuous behaviour of the associated skeleton. This theoretical approach suggests an algorithm for solving the eikonal equation: ‖∇ƒ‖ = g. Finally, we end with some new watershed algorithms, which present the advantage of allowing the use of markers and/or anchor points, thus opening the way towards grey-tone skeletons

A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where a SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin's Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, tracking of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven Sub-Riemannian Geodesics in SE(2)

Detailed and Practical 3D Reconstruction with Advanced Photometric Stereo Modelling

Object 3D reconstruction has always been one of the main objectives of computer vision. After many decades of research, most techniques are still unsuccessful at recovering high resolution surfaces, especially for objects with limited surface texture. Moreover, most shiny materials are particularly hard to reconstruct. Photometric Stereo (PS), which operates by capturing multiple images under changing illumination has traditionally been one of the most successful techniques at recovering a large amount of surface details, by exploiting the relationship between shading and local shape. However, using PS has been highly impractical because most approaches are only applicable in a very controlled lab setting and limited to objects experiencing diffuse reflection. Nevertheless, recent advances in differential modelling have made complicated Photometric Stereo models possible and variational optimisations for these kinds of models show remarkable resilience to real world imperfections such as non-Gaussian noise and other outliers. Thus, a highly accurate, photometric-based reconstruction system is now possible. The contribution of this thesis is threefold. First of all, the Photometric Stereo model is extended in order to be able to deal with arbitrary ambient lighting. This is a step towards acquisition in a non-fully controlled lab setting. Secondly, the need for a priori knowledge of the light source brightness and attenuation characteristics is relaxed as an alternating optimisation procedure is proposed which is able to estimate these parameters. This extension allows for quick acquisition with inexpensive LEDs that exhibit unpredictable illumination characteristics (flickering etc). Finally, a volumetric parameterisation is proposed which allows one to tackle the multi-view Photometric Stereo problem in a similar manner, in a simple unified differential model. This final extension allows for complete object reconstruction merging information from multiple images taken from multiple viewpoints and variable illumination. The theoretical work in this thesis is experimentally evaluated in a number of challenging real world experiments, with data captured by custom-made hardware. In addition, the applicability of the generality of the proposed models is demonstrated by presenting a differential model for the shape of polarisation problem, which leads to a unified optimisation problem, fusing information from both methods. This allows for the acquisition of geometrical information about objects such as semi-transparent glass, hitherto hard to deal with

Variational methods for shape and image registrations.

Estimating and analysis of deformation, either rigid or non-rigid, is an active area of research in various medical imaging and computer vision applications. Its importance stems from the inherent inter- and intra-variability in biological and biomedical object shapes and from the dynamic nature of the scenes usually dealt with in computer vision research. For instance, quantifying the growth of a tumor, recognizing a person\u27s face, tracking a facial expression, or retrieving an object inside a data base require the estimation of some sort of motion or deformation undergone by the object of interest. To solve these problems, and other similar problems, registration comes into play. This is the process of bringing into correspondences two or more data sets. Depending on the application at hand, these data sets can be for instance gray scale/color images or objects\u27 outlines. In the latter case, one talks about shape registration while in the former case, one talks about image/volume registration. In some situations, the combinations of different types of data can be used complementarily to establish point correspondences. One of most important image analysis tools that greatly benefits from the process of registration, and which will be addressed in this dissertation, is the image segmentation. This process consists of localizing objects in images. Several challenges are encountered in image segmentation, including noise, gray scale inhomogeneities, and occlusions. To cope with such issues, the shape information is often incorporated as a statistical model into the segmentation process. Building such statistical models requires a good and accurate shape alignment approach. In addition, segmenting anatomical structures can be accurately solved through the registration of the input data set with a predefined anatomical atlas. Variational approaches for shape/image registration and segmentation have received huge interest in the past few years. Unlike traditional discrete approaches, the variational methods are based on continuous modelling of the input data through the use of Partial Differential Equations (PDE). This brings into benefit the extensive literature on theory and numerical methods proposed to solve PDEs. This dissertation addresses the registration problem from a variational point of view, with more focus on shape registration. First, a novel variational framework for global-to-local shape registration is proposed. The input shapes are implicitly represented through their signed distance maps. A new Sumof- Squared-Differences (SSD) criterion which measures the disparity between the implicit representations of the input shapes, is introduced to recover the global alignment parameters. This new criteria has the advantages over some existing ones in accurately handling scale variations. In addition, the proposed alignment model is less expensive computationally. Complementary to the global registration field, the local deformation field is explicitly established between the two globally aligned shapes, by minimizing a new energy functional. This functional incrementally and simultaneously updates the displacement field while keeping the corresponding implicit representation of the globally warped source shape as close to a signed distance function as possible. This is done under some regularization constraints that enforce the smoothness of the recovered deformations. The overall process leads to a set of coupled set of equations that are simultaneously solved through a gradient descent scheme. Several applications, where the developed tools play a major role, are addressed throughout this dissertation. For instance, some insight is given as to how one can solve the challenging problem of three dimensional face recognition in the presence of facial expressions. Statistical modelling of shapes will be presented as a way of benefiting from the proposed shape registration framework. Second, this dissertation will visit th

A variational technique for three-dimensional reconstruction of local structure

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 1999.Includes bibliographical references (leaves 66-70).by Eric Raphaël Amram.S.M