Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images

We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Our software---CHUNKYEuler---is available as open source on Bitbucket. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams

Computing the Component-Labeling and the Adjacency Tree of a Binary Digital Image in Near Logarithmic-Time

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-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

Algebraic Topology

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook \emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael Grinfeld from the University of Strathclyd

Surface-Based Computation of the Euler Characteristic in the BCC Grid

As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface

On the Topological Disparity Characterization of Square-Pixel Binary Image Data by a Labeled Bipartite Graph

Given an nD digital image I based on cubical n-xel, to fully characterize the degree of internal topological dissimilarity existing in I when using different adjacency relations (mainly, comparing 2n or 2n −1 adjacency relations) is a relevant issue in current problems of digital image processing relative to shape detection or identification. In this paper, we design and implement a new self-dual representation for a binary 2D image I, called {4, 8}-region adjacency forest of I ({4, 8}-RAF, for short), that allows a thorough analysis of the differences between the topology of the 4-regions and that of the 8-regions of I. This model can be straightforwardly obtained from the classical region adjacency tree of I and its binary complement image Ic, by a suitable region label identification. With these two labeled rooted trees, it is possible: (a) to compute Euler number of the set of foreground (resp. background) pixels with regard to 4-adjacency or 8-adjacency; (b) to identify new local and global measures and descriptors of topological dissimilarity not only for one image but also between two or more images. The parallelization of the algorithms to extract and manipulate these structures is complete, thus producing efficient and unsophisticated codes with a theoretical computing time near the logarithm of the width plus the height of an image. Some toy examples serve to explain the representation and some experiments with gray real images shows the influence of the topological dissimilarity when detecting feature regions, like those returned by the MSER (maximally stable extremal regions) method.Ministerio de Economía, Industria y Competitividad PID2019-110455GB-I00 (Par-HoT)Junta de Andalucía US-138107

Face Detection and Recognition using Skin Segmentation and Elastic Bunch Graph Matching

Recently, face detection and recognition is attracting a lot of interest in areas such as network security, content indexing and retrieval, and video compression, because ‘people’ are the object of attention in a lot of video or images. To perform such real-time detection and recognition, novel algorithms are needed, which better current efficiencies and speeds. This project is aimed at developing an efficient algorithm for face detection and recognition. This project is divided into two parts, the detection of a face from a complex environment and the subsequent recognition by comparison. For the detection portion, we present an algorithm based on skin segmentation, morphological operators and template matching. The skin segmentation isolates the face-like regions in a complex image and the following operations of morphology and template matching help reject false matches and extract faces from regions containing multiple faces. For the recognition of the face, we have chosen to use the ‘EGBM’ (Elastic Bunch Graph Matching) algorithm. For identifying faces, this system uses single images out of a database having one image per person. The task is complex because of variation in terms of position, size, expression, and pose. The system decreases this variance by extracting face descriptions in the form of image graphs. In this, the node points (chosen as eyes, nose, lips and chin) are described by sets of wavelet components (called ‘jets’). Image graph extraction is based on an approach called the ‘bunch graph’, which is constructed from a set of sample image graphs. Recognition is based on a directly comparing these graphs. The advantage of this method is in its tolerance to lighting conditions and requirement of less number of images per person in the database for comparison