An entropy-based persistence barcode

In persistent homology, the persistence barcode encodes pairs of simplices meaning birth and death of homology classes. Persistence barcodes depend on the ordering of the simplices (called a filter) of the given simplicial complex. In this paper, we define the notion of “minimal” barcodes in terms of entropy. Starting from a given filtration of a simplicial complex K, an algorithm for computing a “proper” filter (a total ordering of the simplices preserving the partial ordering imposed by the filtration as well as achieving a persistence barcode with small entropy) is detailed, by way of computation, and subsequent modification, of maximum matchings on subgraphs of the Hasse diagram associated to K. Examples demonstrating the utility of computing such a proper ordering on the simplices are given

Effects of Correlated Noise on the Performance of Persistence Based Dynamic State Detection Methods

The ability to characterize the state of dynamic systems has been a pertinent task in the time series analysis community. Traditional measures such as Lyapunov exponents are often times difficult to recover from noisy data, especially if the dimensionality of the system is not known. More recent binary and network based testing methods have delivered promising results for unknown deterministic systems, however noise injected into a periodic signal leads to false positives. Recently, we showed the advantage of using persistent homology as a tool for achieving dynamic state detection for systems with no known model and showed its robustness to white Gaussian noise. In this work, we explore the robustness of the persistence based methods to the influence of colored noise and show that colored noise processes of the form $1/f^{\alpha}$ lead to false positive diagnostic at lower signal to noise ratios for $\alpha<0$

Towards personalized diagnosis of Glioblastoma in Fluid-attenuated inversion recovery (FLAIR) by topological interpretable machine learning

Glioblastoma multiforme (GBM) is a fast-growing and highly invasive brain tumour, it tends to occur in adults between the ages of 45 and 70 and it accounts for 52 percent of all primary brain tumours. Usually, GBMs are detected by magnetic resonance images (MRI). Among MRI, Fluid-attenuated inversion recovery (FLAIR) sequence produces high quality digital tumour representation. Fast detection and segmentation techniques are needed for overcoming subjective medical doctors (MDs) judgment. In the present investigation, we intend to demonstrate by means of numerical experiments that topological features combined with textural features can be enrolled for GBM analysis and morphological characterization on FLAIR. To this extent, we have performed three numerical experiments. In the first experiment, Topological Data Analysis (TDA) of a simplified 2D tumour growth mathematical model had allowed to understand the bio-chemical conditions that facilitate tumour growth: the higher the concentration of chemical nutrients the more virulent the process. In the second experiment topological data analysis was used for evaluating GBM temporal progression on FLAIR recorded within 90 days following treatment (e.g., chemo-radiation therapy - CRT) completion and at progression. The experiment had confirmed that persistent entropy is a viable statistics for monitoring GBM evolution during the follow-up period. In the third experiment we had developed a novel methodology based on topological and textural features and automatic interpretable machine learning for automatic GBM classification on FLAIR. The algorithm reached a classification accuracy up to the 97%.Comment: 22 pages; 16 figure