Karol Szustakowski

Titouan Le Breton, Karol Szustakowski, Marie Piraud

We present Flash Cubical, a highly efficient computation of cubical persistence on a V-filtration for 2D and 3D images over $\mathbb{F}_2$. The implementation is built around three core ideas. First, cubical complexes satisfy properties that allow for the computation of persistence of the highest dimension via union-find and duality. Second, pruning of certain edges allows for a fast and efficient implementation of union-find. Third, the use of a lookup table, which exploits the regularity of cubical complexes to pre-compute local information. This avoids the need to compute local information at run time. To the best of our knowledge, this is the most efficient implementation of cubical persistence with a V-filtration, both in terms of time and memory costs. Although the paper focuses on persistence for V-filtration cubical complexes, the underlying ideas generalise naturally to T-filtrations on cubical complexes and suggest promising directions for other complexes.

Karol Szustakowski, Luk Frank, Julia Esser et al.

In many biomedical segmentation tasks, the preservation of elongated structure continuity and length is more important than voxel-wise accuracy. We propose two novel loss functions, Negative Centerline Loss and Simplified Topology Loss, that, applied to Convolutional Neural Networks (CNNs), help preserve connectivity of output instances. Moreover, we discuss characteristics of experiment design, such as downscaling and spacing correction, that help obtain continuous segmentation masks. We evaluate our approach on a 3D light-sheet fluorescence microscopy dataset of axon initial segments (AIS), a task prone to discontinuity due to signal dropout. Compared to standard CNNs and existing topology-aware losses, our methods reduce the number of segmentation discontinuities per instance, particularly in regions with missing input signal, resulting in improved instance length calculation in downstream applications. Our findings demonstrate that structural priors embedded in the loss design can significantly enhance the reliability of segmentation for biological applications.