Minimal simple pairs in the cubic grid

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs

Cascaded multitask U-Net using topological loss for vessel segmentation and centerline extraction

Vessel segmentation and centerline extraction are two crucial preliminary tasks for many computer-aided diagnosis tools dealing with vascular diseases. Recently, deep-learning based methods have been widely applied to these tasks. However, classic deep-learning approaches struggle to capture the complex geometry and specific topology of vascular networks, which is of the utmost importance in most applications. To overcome these limitations, the clDice loss, a topological loss that focuses on the vessel centerlines, has been recently proposed. This loss requires computing, with a proposed soft-skeleton algorithm, the skeletons of both the ground truth and the predicted segmentation. However, the soft-skeleton algorithm provides suboptimal results on 3D images, which makes the clDice hardly suitable on 3D images. In this paper, we propose to replace the soft-skeleton algorithm by a U-Net which computes the vascular skeleton directly from the segmentation. We show that our method provides more accurate skeletons than the soft-skeleton algorithm. We then build upon this network a cascaded U-Net trained with the clDice loss to embed topological constraints during the segmentation. The resulting model is able to predict both the vessel segmentation and centerlines with a more accurate topology.Comment: 13 pages, 4 figure

Topological monsters in Z^3: A non-exhaustive bestiary

International audienceSimple points in Z^n, and especially in Z^3, are the basis of several topology-preserving transformation methods proposed for image analysis (segmentation, skeletonisation, ...). Most of these methods rely on the assumption that the --iterative or parallel-- removal of simple points from a discrete object X necessarily leads to a globally minimal topologically equivalent sub-object of X (i.e. a subset Y which is topologically equivalent to X and which does not strictly include another set Z topologically equivalent to X). This is however false in Z^3, and more generally in Z^n. We illustrate this fact by presenting some topological monsters, i.e. some objects of Z^3 only composed of non-simple points, but which could however be reduced without altering their topology

Combinatorial structure of rigid transformations in 2D digital images

International audienceRigid transformations are involved in a wide range of digital image processing applications. When applied on such discrete images, rigid transformations are however usually performed in their associated continuous space, then requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as a fully discrete process. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on any subset of Z^2 of size N*N. We describe this combinatorial structure, which presents a space complexity O(N^9) and we propose an algorithm enabling to build it in linear time with respect to this space complexity. This algorithm, which handles real (i.e. non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation

Topological properties of thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. In contrast to the situation in Z^n , n>=3, it was proved in the 80s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds

Digital imaging: A unified topological framework

International audienceIn this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on Z^n and equipped with a standard pair of adjacencies) as an image defined in the space of cubical complexes (F^n). In particular, it is shown that all the standard pairs of adjacencies (namely the (4, 8) and (8, 4)-adjacencies in Z^2, the (6, 18), (18, 6), (6, 26), and (26, 6)-adjacencies in Z^3 , and more generally the (2n, 3n−1) and (3n−1, 2n)-adjacencies in Z^n) can then be correctly modelled in F^n . Moreover, it is established that the digital fundamental group of a digital image in Z^n is isomorphic to the fundamental group of its corresponding image in F^n , thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (Z^n) or cubical complexes (F^n), and to potentially unify some of them

Topology-preserving thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. Z^n, n ≥ 3), it was proved in the 80’s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to more general spaces (the 2-D pseudomanifolds) and objects (the 2-D cubical complexes)

Direction-adaptive grey-level morphology. Application to 3D vascular brain imaging

International audienceSegmentation and analysis of blood vessels is an important issue in medical imaging. In 3D cerebral angiographic data, the vascular signal is however hard to accurately detect and can, in particular, be disconnected. In this article, we present a procedure utilising both linear, Hessian-based and morphological methods for blood vessel edge enhancement and reconnection. More specifically, multi-scale second-order derivative analysis is performed to detect candidate vessels as well as their orientation. This information is then fed to a spatially-variant morphological filter for reconnection and reconstruction. The result is a fast and effective vessel-reconnecting method

Bijective rigid motions of the 2D Cartesian grid

International audienceRigid motions are fundamental operations in image processing. While they are bijective and isometric in R^2, they lose these properties when digitized in Z^2. To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on Z^2, initially proposed by Nouvel and Rémila for rotations on Z^2. This allows us to study bijective rigid motions on Z^2, and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of Z^2 is bijective

Combinatorial properties of 2D discrete rigid transformations under pixel-invariance constraints

International audienceRigid transformations are useful in a wide range of digital image processing applications. In this context, they are generally considered as continuous processes, followed by discretization of the results. In recent works, rigid transformations on ℤ^2 have been formulated as a fully discrete process. Following this paradigm, we investigate --from a combinatorial point of view-- the effects of pixel-invariance constraints on such transformations. In particular we describe the impact of these constraints on both the combinatorial structure of the transformation space and the algorithm leading to its generation