Leveraging the geodesic distance transform on interval-valued maps to compute the tree of shapes

The tree of shapes (ToS) is a self-dual and contrast-invariant hierarchical representation of images, making it highly suitable for various image processing tasks such as filtering, segmentation, and object detection. Computing the ToS has been shown to be linear-time under certain conditions, but encounters challenges when applied to high-dynamic range (HDR) images or suffers from scalability issues on massively parallel architectures. This efficient algorithm relies on a specific topological framework that allows continuous topological properties on discrete images. Interval-valued maps ensure the correctness of the ToS construction by providing a continuous representation of the image. In this paper, we introduce geodesics on interval-valued maps and leverage the resulting distance maps to compute the ToS. By utilizing the geodesic distance transform, we can efficiently compute the ToS on high-dynamic range images and overcome the scalability issues associated with existing algorithms. The proposed approach allows for a more efficient computation of the ToS, making it suitable for processing large-scale images and enabling its application in various image processing tasks.