Structural Analysis of the Additive Noise Impact on the -tree

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 simplification, object detection, or segmentation. However, it has been demonstrated that these tasks are very sensitive to the presence of noise in images. While the quality of some alpha-tree applications has been studied, including some with noisy images, the noise impact on the whole structure has been little investigated. Thus, in this paper, we examine the structure of alpha-trees built on images in the presence of noise with respect 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. We finally observe that the choice of the energy has a great impact on the tree structure.