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Taking advantage of the topological and isotopic properties of binary digital images, we present here anew algorithm for connected component labeling (CLL). A local-to-global treatment of the topologicalinformation within the image, allows us to develop an inherent parallel approach. The time complexityorder for an image of m ×n pixels, under the assumption that a processing element exists for each pixel, is near O (log(m + n )) . Additionally, our method computes both the foreground and background CCL, and allows a straightforward computation of topological features like Adjacency Trees. Experiments show thatour method obtains better performance metrics than other approaches. Our work aims at generating anew class of labeling algorithms: those centered in fully parallel approaches based on computationaltopology, thus allowing a perfect concurrent execution in multiple threads and preventing the use ofcritical sections and atomic instructions. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

International audience; Discrete gradient vector fields are combinatorial structures that can be used for accelerating the homology computation of CW complexes, such as simplicial or cubical complexes, by reducing their number of cells. Consequently, they provide a bound for the Betti numbers (the most basic homological information). A discrete gradient vector field can eventually reduce the complex to its minimal form, having as many cells of each dimension as its corresponding Betti number, but this is not guaranteed. Moreover, finding an optimal discrete gradient vector field is an NP-hard problem. We describe here a generalization, which we call Homological Discrete Vector Field (HDVF), which can overcome these limitations by allowing cycles under a certain algebraic condition. In this work we define the HDVF and its associated reduction, we study how to efficiently compute a HDVF, we establish the relation between the HDVF and other concepts in computational homology and we estimate the average complexity of its computation. We also introduce five basic operations for modifying a HDVF, which can also be applied to discrete gradient vector fields.

In this paper we develop a new computational technique called boundary scale-space theory. This tech- nique is based on the topol1 ogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects { Ki }i ≥ 1 all of them having the same number of closed pieces than K. Each piece of Ki ( ∀i ≥ 1) presents the same interior part than the corresponding one in K, and a different boundary part depending on the scale i. Working with coefficients in a field, a scale is installed for the algebraic boundary of each piece and a new invariant for cell complex isomorphisms is given in terms of the Betti numbers of the generated boundary-scale-space cell complexes. Moreover, the so called homology boundary scale-space model of K ( hbss -model for short) is introduced here. Thismodel consists of a hierarchical graph whose nodes are the homology generators of the different bound- ary scale levels and whose edges are specified by homology generators of consecutive boundary scaleindices linked by ( hbss -transition maps) preserving homology classes. Various codes for each connectedsubgraph of an hbss -model are defined, which besides being fast and efficient similarity measures for cel- lular structures, they are as well relevant interpretive tools for the hbss -model. Finally, experimentations mainly aimed at clarifying and understanding the notion of hbss -model, as well as conjecturing about new graph isomorphism invariants (seeing graphs as a 1-dimensional cell complexes), are performed. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2016-77785-P

In [14], a topologically consistent framework to support parallel topological analysis and recognition for 2 D digital objects was introduced. Based on this theoretical work, we focus on the problem of finding efficient algorithmic solutions for topological interrogation of a 2 D digital object of interest D of a pre-segmented digital image I , using 4-adjacency between pixels of D . In order to maximize the degree of parallelization of the topological processes, we use as many elementary unit processing as pixels the image I has. The mathematical model underlying this framework is an appropriate extension of the classical concept of abstract cell complex: a primal–dual abstract cell complex (pACC for short). This versatile data structure encompasses the notion of Homological Spanning Forest fostered in [14,15]. Starting from a symmetric pACC associated with I , the modus operandi is to construct via combinatorial operations another asymmetric one presenting the maximal number of non-null primal elementary interactions between the cells of D . The fundamental topological tools have been transformed so as to promote an efficient parallel implementation in any parallel-oriented architecture (GPUs, multi-threaded computers, SIMD kernels and so on). A software prototype modeling such a parallel framework is built.

Segmentations of a digital object based on a connectivity criterion at n-xel or sub-n-xel level are useful tools in image topological analysis and recognition. Working with cell complex analogous of digital objects, an example of this kind of segmentation is that obtained from the combinatorial representation so called Homological Spanning Forest (HSF, for short) which, informally, classifies the cells of the complex as belonging to regions containing the maximal number of cells sharing the same homological (algebraic homology with coefficient in a field) information. We design here a parallel method for computing a HSF (using homology with coefficients in Z/2Z) of a 3D digital object. If this object is included in a 3D image of m1 × m2 × m3 voxels, its theoretical time complexity order is near O(log(m1 + m2 + m3)), under the assumption that a processing element is available for each voxel. A prototype implementation validating our results has been written and several synthetic, random and medical tridimensional images have been used for testing. The experiments allow us to assert that the number of iterations in which the homological information is found varies only to a small extent from the theoretical computational time. Ministerio de Economía y Competitividad MTM2016-81030-P

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

In this paper, the concept of homogeneity is defined, from a topological perspective, in order to analyze how uniform is the material composition in 2D electron microscopy images. Topological multiresolution parameters are taken into account to obtain better results than classical techniques. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

Given a 3D binary digital image I, we define and compute an edge-weighted tree, called Homological Region Tree (or Hom-Tree, for short). It coincides, as unweighted graph, with the classical Region Adjacency Tree of black 6-connected components (CCs) and white 26- connected components of I. In addition, we define the weight of an edge (R, S) as the number of tunnels that the CCs R and S “share”. The Hom-Tree structure is still an isotopic invariant of I. Thus, it provides information about how the different homology groups interact between them, while preserving the duality of black and white CCs. An experimentation with a set of synthetic images showing different shapes and different complexity of connected component nesting is performed for numerically validating the method. Ministerio de Economía y Competitividad MTM2016-81030-P

Connected component labeling (CCL) of binary images is one of the fundamental operations in real time applications. The adjacency tree (AdjT) of the connected components offers a region-based representation where each node represents a region which is surrounded by another region of the opposite color. In this paper, a fully parallel algorithm for computing the CCL and AdjT of a binary digital image is described and implemented, without the need of using any geometric information. The time complexity order for an image of m × n pixels under the assumption that a processing element exists for each pixel is near O(log(m+ n)). Results for a multicore processor show a very good scalability until the so-called memory bandwidth bottleneck is reached. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

In this paper, a homotopy-based method is employed for the recovery of speech recordings from missing or corrupted samples taken in a noisy environment. The model for the acquisition device is a compressed sensing scenario using Gabor frames. To recover an approximation of the speech file, we used the basis pursuit denoising method with the homotopy continuation algorithm. We tested the proposed method with various speech recordings.

This is an experimental paper in which we introduce the possibility to analyze and to synthesize 3D medical images by using multivariate Gabor frames with Gaussian windows. Our purpose is to apply a space-variant filter-like operation in the space-frequency domain to correct medical images corrupted by different types of acquisitions errors. The Gabor frames are constructed with Gaussian windows sampled on non-separable lattices for a better packing of the space-frequency plane. An implementable solution for 3D-Gabor frames with non-separable lattice is given and numerical tests on simulated data are presented. Austrian Science Fund (FWF) P27516

An appropriate generalization of the classical notion of abstract cell complex, called primal-dual abstract cell complex (pACC for short) is the combinatorial notion used here for modeling and analyzing the topology of nD digital objects and images. Let D ⊂ I be a set of n-xels (ROI) and I be a n-dimensional digital image.We design a theoretical parallel algorithm for constructing a topologically meaningful asymmetric pACC HSF(D), called Homological Spanning Forest of D (HSF of D, for short) starting from a canonical symmetric pACC associated to I and based on the application of elementary homotopy operations to activate the pACC processing units. From this HSF-graph representation of D, it is possible to derive complete homology and homotopy information of it. The preprocessing procedure of computing HSF(I) is thoroughly discussed. In this way, a significant advance in understanding how the efficient HSF framework for parallel topological computation of 2D digital images developed in [2] can be generalized to higher dimension is made. Ministerio de Economía y Competitividad TEC2016-77785-P Ministerio de Economía y Competitividad MTM2016-81030-P

In [18], a membrane parallel theoretical framework for computing (co)homology information of foreground or background of binary digital images is developed. Starting from this work, we progress here in two senses: (a) providing advanced topological information, such as (co)homology torsion and efficiently answering to any decision or classification problem for sum of k-xels related to be a (co)cycle or a (co)boundary; (b) optimizing the previous framework to be implemented in using GPGPU computing. Discrete Morse theory, Effective Homology Theory and parallel computing techniques are suitably combined for obtaining a homological encoding, called algebraic minimal model, of a Region-Of-Interest (seen as cubical complex) of a presegmented k-D digital image.

Given a binary object (2D or 3D), its Betti numbers characterize the number of holes in each dimension. They are obtained algebraically, and even though they are perfectly defined, there is no unique way to display these holes. We propose two geometric measures for the holes, which are uniquely defined and try to compensate the loss of geometric information during the homology computation: the thickness and the breadth. They are obtained by filtering the information of the persistent homology computation of a filtration defined through the signed distance transform of the binary object.

n-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. Further approaches deal mainly with a pre-processing of the matrices in order to reduce their size. In this paper we propose a new approach, initially based on discrete Morse theory, which computes some homological information (Betti numbers and representative cycles) without calculating the Normal Smith Form. It works on any dimension, and it can also be applied to any kind of regular cell complex.

In this paper, we follow up on the studies developed by Kovalevsky (Comput Vis Graph Image Process 46:141---161, 1989) and Kenmochi et al. (Comput Vis Image Underst 71:281---293, 1998), which defined a continuous analog for a 4-dimensional digital object. Here, we construct a cell complex that has the same topological information as the original 4-dimensional digital object.

In this paper, we introduce a simple and original algorithm to compute a three-dimensional simplicial complex topologically equivalent to a 3D digital object V, according to the 26-adjacency. The use of this adjacency generates issues like auto-intersecting triangles that unnecessarily increase the dimensionality of the associated simplicial complex. To avoid these problems, we present an approach based on a modified Delaunay tetrahedralization of the digital object, that preserves its topological characteristics. Considering the resulting complex as an input in algebraic-topological format (fixing a ground ring for the coefficients), we develop propositions regardless of the adjacency considered. These potential applications are related to topological analysis like thinning, homology computation, topological characterization and control. Moreover, our technique is susceptible to be extended to higher dimensions.

Poster session: Biometrics; International audience; Finding facial features respectively under expression and illumination variations is always a difficult problem. One popular solution for improving the performance of facial point localization is to use the spatial relation between facial feature positions. While existing algorithms mostly rely on the priori knowledge of facial structure and on a training phase, this paper presents an online approach without requirements of pre-defined constraints on feature distributions. Instead of training specific detectors for each facial feature, a generic method is first used to extract a set of interest points from test images. With a robust feature descriptor named Patterns Oriented Edge Magnitude (POEM), a smaller set of these points are picked as candidates. Then we apply a game-theoretic technique to select facial points from the candidates, while the global geometric properties of face are well preserved. The experimental results demonstrate that our method achieves satisfactory performance for face images under expression and lighting variations

International audience; We address the problem of jointly estimating the scene illumination, the radiometric camera calibration and the reflectance properties of an object using a set of images from a community photo collection. The highly ill-posed nature of this problem is circumvented by using appropriate representations of illumination, an empirical model for the nonlinear function that relates image irradiance with intensity values and additional assumptions on the surface reflectance properties. Using a 3D model recovered from an unstructured set of images, we estimate the coefficients that represent the illumination for each image using a frequency framework. For each image, we also compute the correspond- ing camera response function. Additionally, we calculate a simple model for the reflectance properties of the 3D model. A robust non-linear optimization is proposed exploiting the high sparsity present in the problem.

International audience; This short paper accompanies an invited lecture on a historical survey of geometric computer vision problems. It presents some early works on image-based 3D modeling, multi-view geometry, and structurefrom- motion, from the last three centuries. Some of these are relatively well known to photogrammetrists and computer vision researchers whereas others seem to have been largely forgotten or overlooked. This paper gives a very brief summary of an ongoing historical study.