Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Abstract

Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.

On the effectiveness of persistent homology

“…Another way to give more weight to short intervals, or intervals of any persistence, is given by an appropriate choice of the weighing function for persistence images. Note, however, that the stability results also depend on the choice of filtration, persistence signature and metric [84]. Finally, we note that there has recently been a lot of effort in trying to train neural networks to learn what the best PH signature is for specific types of applications [19,36,59,74].…”

“…The steps for extracting PH features are visualized in Figure 1. For good choices of filtration and signature [84], there are theoretical results that guarantee that PH is stable under small perturbations [23,80]. After the PH signature is calculated, statistical hypothesis testing [10,15], or machine learning techniques such as SVM or k-NN [2,40,62] can be used on these features to study the differences within the dataset of interest.…”

“…If the data is noisy, it is better to replace the distance function with the Distanceto-Measure (DTM) δ X,m : R n → R, where δ X,m (x) is the average distance from a number of neighbors on the point cloud [5,24] (see Figure 9). However, depending on the task, there are many other filtration functions one could choose, such as rank [67], height, radial, erosion, dilation [40], and the resulting PH captures completely different information about the cycles [84]. For example, whereas PH with respect to the Vietoris-Rips filtration encodes the size of the hole (Figure 1), PH on the height filtration informs about the position of the hole (Figure 15).…”

Topology-based fluorescence image analysis for automated cell identification and segmentation

“…Computationally, the primary segmentation's runtime is on-par with existing machine learning methods, but preprocessing and boundary identification often take longer [6,9]. While TOBLERONE can function with only a single parameter, it is not always obvious how to appropriately select its value -however, topological features in images are typically robust to variations in intensity, so a range of persistence thresholds will usually return a suitable segmentation [26][27][28]. As a topological image analysis technique, TOBLERONE is invariant of cell and organelle morphology [29,30].…”

Feature detection and hypothesis testing for extremely noisy nanoparticle images using topological data analysis

“…The GUDHI library considers an elementary cube τ to have the minimum value of the all the d-cubes ν containing τ . We consider the homology of sublevel set filtrations 1 , so black pixel values 1 Other filtrations could be considered here; however, besides the sublevel set filtration, all require choosing a threshold at which to binarize the image (Turkes et al, 2021;Garin and Tauzin, 2019)-see also the opening paragraph of Section 4.…”