The rapid evolution of image processing equipment and techniques ensures the development of novel picture analysis methodologies. One of the most powerful yet computationally possible algebraic techniques for measuring the topological characteristics of functions is persistent homology. It's an algebraic invariant that can capture topological details at different spatial resolutions. Persistent homology investigates the topological features of a space using a set of sampled points, such as pixels. It can track the appearance and disappearance of topological features caused by changes in the nested space created by an operation known as filtration, in which a parameter scale, in our case the intensity of pixels, is increased to detect changes in the studied space over a range of varying scales. In addition, at the level of machine learning there were many studies and articles witnessing recently the combination between homological persistence and machine learning algorithms. On another level, prostate cancer is diagnosed referring to a scoring criterion describing the severity of the cancer called Gleason score. The classical Gleason system defines five histological growth patterns (grades). In our study we propose to study the Gleason score on some glands issued from a new optical microscopy technique called SLIM. This new optical microscopy technique that combines two classic ideas in light imaging: Zernike’s phase contrast microscopy and Gabor’s holography. Persistent homology features are computed on these images. We suggested machine learning methods to classify these images into the corresponding Gleason score. Machine learning techniques applied on homological persistence features was very effective in the detection of the right Gleason score of the prostate cancer in these kinds of images and showed an accuracy of above 95%.

Models for noncoherent error control in random linear network coding (RLNC) and store and forward (SAF) have been recently proposed. In this paper, we model different types of random network communications as the transmission of flats of matroids. This novel framework encompasses RLNC and SAF and allows us to introduce a novel protocol, referred to as random affine network coding (RANC), based on affine combinations of packets. Although the models previously proposed for RLNC and SAF only consider error control, using our framework, we first evaluate and compare the performance of different network protocols in the error-free case. We define and determine the rate, average delay, and throughput of such protocols, and we also investigate the possibilities of partial decoding before the entire message is received. We thus show that RANC outperforms RLNC in terms of data rate and throughput thanks to a more efficient encoding of messages into packets. Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for RLNC and SAF. Error control is thus reduced to a coding-theoretic problem on flats of a matroid, where two distinct metrics can be used for error correction. We study the maximum cardinality of codes on flats in general, and codes for error correction in RANC in particular. We finally design a class of nearly optimal codes for RANC based on rank metric codes for which we propose a low-complexity decoding algorithm. The gain of RANC over RLNC is thus preserved with no additional cost in terms of complexity.

Spectral graph analysis based on Laplace operators has been ubiquitously applied in graph signal processing. Extending these tools to a generalization of graphs is helpful for several applications, such as network analysis, sensor network coverage problem and other fields where the relationships between two or more vertices should be modeled. Such mathematical theories already exist, but the associated algorithms are still in their infancy. In this article, we propose a new algorithm to compute harmonic forms, i.e. solutions of the Laplace equation, whose implementation is simple enough to be distributed among a network without central control center.

Homology groups are a prime tool for measuring the connectivity of a network, and their computation in a distributed and adaptive way is mandatory for their use in sensor networks. In this paper, we propose a solution based on the construction of an adaptive discrete vector field from where, thanks to the discrete Morse theory, the generators of the homology groups are extracted. The efficiency and the adaptability of our approach are tested against two applications: the detection and the localization of the holes in the coverage, and the selection of active sensors ensuring complete coverage.

Impulsive noise, where large amplitudes arise with a relatively high probability, arises in many communication systems including interference in Low Power Wide Area Networks. A challenge in coping with impulsive noise, particularly alpha-stable models, is that tractable expressions for the log-likelihood ratio (LLR) are not available, which has a large impact on soft-input decoding schemes, e.g., low-density parity-check (LDPC) packets. On the other hand, constraints on packet length also mean that pilot signals are not available resulting in non-trivial approximation and parameter estimation problems for the LLR. In this paper, a new unsupervised parameter estimation algorithm is proposed for LLR approximation. In terms of the frame error rate (FER), this algorithm is shown to significantly outperform existing unsupervised estimation methods for short LDPC packets (on the order of 500 symbols), with nearly the same performance as when the parameters are perfectly known. The performance is also compared with an upper bound on the information-theoretic limit for the FER, which suggests that in impulsive noise further improvements require the use of an alternative code structure other than LDPC.