Abstract
Hierarchical representations are very convenient tools when working with images. Among them, the $\alpha$-tree is the basis of several powerful hierarchies used for various applications such as image simplifi- cation, object detection, or segmentation. However, it has been demon- strated that these tasks are very sensitive to the noise corrupting the image. While the quality of some $\alpha$-tree applications has been studied, including some with noisy images, the noise impact on the whole struc- ture has been little investigated. Thus, in this paper, we examine the structure of $\alpha$-trees built on images corrupted by some noise with re- spect to the noise level. We compare its effects on constant and natural images, with different kinds of content, and we demonstrate the relation between the noise level and the distribution of every $\alpha$-tree node depth. Furthermore, we extend this study to the node persistence under a given energy criterion, and we propose a novel energy definition that allows assessing the robustness of a region to the noise.