  Any image is perceived subconsciously as a coherent structure (or whole) with
two contrast substructures: figure and ground. The figure consists of numerous
auto-generated substructures with an inherent hierarchy of far more smalls than
larges. Through these substructures, the structural beauty of an image (L) can
be computed by the multiplication of the number of substructures (S) and their
inherent hierarchy (H). This definition implies that the more substructures,
the more living or more structurally beautiful, and the higher hierarchy of the
substructures, the more living or more structurally beautiful. This is the
non-recursive approach to the structural beauty of images or the livingness of
space. In this paper we develop a recursive approach, which derives all
substructures of an image (instead of its figure) and continues the deriving
process for those decomposable substructures until none of them are
decomposable. All of the substructures derived at different iterations (or
recursive levels) together constitute a living structure; hence the notion of
living images. We applied the recursive approach to a set of images and found
that (1) the number of substructures of an image is far lower (3 percent on
average) than the number of pixels and the centroids of the substructures can
effectively capture the skeleton or saliency of the image; (2) all the images
have the recursive levels more than three, indicating that they are indeed
living images; (3) no more than 2 percent of the substructures are
decomposable; (4) structural beauty can be measured by the recursively defined
substructures, as well as their decomposable subsets. The recursive approach is
proved to be more robust than the non-recursive approach. The recursive
approach and the non-recursive approach both provide a powerful means to study
the livingness or vitality of space in cities and communities.
