Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model

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-

Enhanced Parallel Generation of Tree Structures for the Recognition of 3D Images

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-

Toward Parallel Computation of Dense Homotopy Skeletons for nD Digital Objects

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-PMinisterio de Economía y Competitividad MTM2016-81030-

Labeling Color 2D Digital Images in Theoretical Near Logarithmic Time

A design of a parallel algorithm for labeling color flat zones (precisely, 4-connected components) of a gray-level or color 2D digital image is given. The technique is based in the construction of a particular Homological Spanning Forest (HSF) structure for encoding topological information of any image.HSFis a pair of rooted trees connecting the image elements at inter-pixel level without redundancy. In order to achieve a correct color zone labeling, our proposal here is to correctly building a sub- HSF structure for each image connected component, modifying an initial HSF of the whole image. For validating the correctness of our algorithm, an implementation in OCTAVE/MATLAB is written and its results are checked. Several kinds of images are tested to compute the number of iterations in which the theoretical computing time differs from the logarithm of the width plus the height of an image. Finally, real images are to be computed faster than random images using our approach.Ministerio de Economía y Competitividad TEC2016-77785-PMinisterio de Economía y Competitividad MTM2016-81030-

Parallel Image Processing Using a Pure Topological Framework

Image processing is a fundamental operation in many real time applications, where lots of parallelism can be extracted. Segmenting the image into different connected components is the most known operations, but there are many others like extracting the region adjacency graph (RAG) of these regions, or searching for features points, being invariant to rotations, scales, brilliant changes, etc. Most of these algorithms part from the basis of Tracing-type approaches or scan/raster methods. This fact necessarily implies a data dependence between the processing of one pixel and the previous one, which prevents using a pure parallel approach. In terms of time complexity, this means that linear order O(N) (N being the number of pixels) cannot be cut down. In this paper, we describe a novel approach based on the building of a pure Topological framework, which allows to implement fully parallel algorithms. Concerning topological analysis, a first stage is computed in parallel for every pixel, thus conveying the local neighboring conditions. Then, they are extended in a second parallel stage to the necessary global relations (e.g. to join all the pixels of a connected component). This combinatorial optimization process can be seen as the compression of the whole image to just one pixel. Using this final representation, every region can be related with the rest, which yields to pure topological construction of other image operations. Besides, complex data structures can be avoided: all the processing can be done using matrixes (with the same indexation as the original image) and element-wise operations. The time complexity order of our topological approach for a m×n pixel image is near O(log(m+n)), under the assumption that a processing element exists for each pixel. Results for a multicore processor show very good scalability until the memory bandwidth bottleneck is reached, both for bigger images and for much optimized implementations. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures.1Ministerio de Economía y Competitividad (España) TEC2012-37868-C04-02AEI/FEDER (UE) MTM2016-81030-PVPPI of the University of Sevill

A parallel Homological Spanning Forest framework for 2D topological image analysis

In [14], a topologically consistent framework to support parallel topological analysis and recognition for2 D digital objects was introduced. Based on this theoretical work, we focus on the problem of findingefficient 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 ofparallelization of the topological processes, we use as many elementary unit processing as pixels theimage I has. The mathematical model underlying this framework is an appropriate extension of the clas- sical concept of abstract cell complex: a primal–dual abstract cell complex (pACC for short). This versatiledata structure encompasses the notion of Homological Spanning Forest fostered in [14,15]. Starting froma symmetric pACC associated with I , the modus operandi is to construct via combinatorial operationsanother asymmetric one presenting the maximal number of non-null primal elementary interactions be- tween the cells of D . The fundamental topological tools have been transformed so as to promote anefficient 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.Ministerio de Educación y Ciencia TEC2012-37868-C04-02/0

Homological Spanning Forests for Discrete Objects

Computing and representing topological information form an important part in many applications such as image representation and compression, classification, pattern recognition, geometric modelling, etc. The homology of digital objects is an algebraic notion that provides a concise description of their topology in terms of connected components, tunnels and cavities. The purpose of this work is to develop a theoretical and practical frame- work for efficiently extracting and exploiting useful homological information in the context of nD digital images. To achieve this goal, we intend to combine known techniques in algebraic topology, and image processing. The main notion created for this purpose consists of a combinatorial representation called Homological Spanning Forest (or HSF, for short) of a digital object or a digital image. This new model is composed of a set of directed forests, which can be constructed under an underlying cell complex format of the image. HSF’s are based on the algebraic concept of chain homotopies and they can be considered as a suitable generalization to higher dimensional cell complexes of the topological meaning of a spanning tree of a geometric graph. Based on the HSF representation, we present here a 2D homology-based framework for sequential and parallel digital image processing.Premio Extraordinario de Doctorado U

Homological spanning forest framework for 2D image analysis

A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions

Exploiting Spatio-Temporal Coherence for Video Object Detection in Robotics

This paper proposes a method to enhance video object detection for indoor environments in robotics. Concretely, it exploits knowledge about the camera motion between frames to propagate previously detected objects to successive frames. The proposal is rooted in the concepts of planar homography to propose regions of interest where to find objects, and recursive Bayesian filtering to integrate observations over time. The proposal is evaluated on six virtual, indoor environments, accounting for the detection of nine object classes over a total of ∼ 7k frames. Results show that our proposal improves the recall and the F1-score by a factor of 1.41 and 1.27, respectively, as well as it achieves a significant reduction of the object categorization entropy (58.8%) when compared to a two-stage video object detection method used as baseline, at the cost of small time overheads (120 ms) and precision loss (0.92).</p

Parallel homological calculus for 3D binary digital images

Topological representations of binary digital images usually take into consideration different adjacency types between colors.Within the cubical-voxel 3D binary image context, we design an algorithm for computing the isotopic model of an image, called (6, 26)-Homological Region Adjacency Tree ((6, 26)-Hom-Tree). This algorithm is based on a flexible graph scaffolding at the inter-voxel level called Homological Spanning Forest model (HSF). HomTrees are edge-weighted trees in which each node is a maximally connected set of constantvalue voxels, which is interpreted as a subtree of the HSF. This representation integrates and relates the homological information (connected components, tunnels and cavities) of the maximally connected regions of constant color using 6-adjacency and 26-adjacency for black and white voxels, respectively (the criteria most commonly used for 3D images). The EulerPoincaré numbers (which may as well be computed by counting the number of cells of each dimension on a cubical complex) and the connected component labeling of the foreground and background of a given image can also be straightforwardly computed from its Hom-Trees. Being ID a 3D binary well-composed image (where D is the set of black voxels), an almost fully parallel algorithm for constructing the Hom-Tree via HSF computation is implemented and tested here. If ID has m1×m2×m3 voxels, the time complexity order of the reproducible algorithm is near O(log(m1+m2+m3)), under the assumption that a processing element is available for each cubical voxel. Strategies for using the compressed information of the Hom-Tree representation to distinguish two topologically different images having the same homological information (Betti numbers) are discussed here. The topological discriminatory power of the Hom-Tree and the low time complexity order of the proposed implementation guarantee its usability within machine learning methods for the classification and comparison of natural 3D images