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Any image is perceived subconsciously as a coherent structure (or whole) withtwo contrast substructures: figure and ground. The figure consists of numerousauto-generated substructures with an inherent hierarchy of far more smalls thanlarges. Through these substructures, the structural beauty of an image (L) canbe computed by the multiplication of the number of substructures (S) and theirinherent hierarchy (H). This definition implies that the more substructures,the more living or more structurally beautiful, and the higher hierarchy of thesubstructures, the more living or more structurally beautiful. This is thenon-recursive approach to the structural beauty of images or the livingness ofspace. In this paper we develop a recursive approach, which derives allsubstructures of an image (instead of its figure) and continues the derivingprocess for those decomposable substructures until none of them aredecomposable. All of the substructures derived at different iterations (orrecursive levels) together constitute a living structure; hence the notion ofliving images. We applied the recursive approach to a set of images and foundthat (1) the number of substructures of an image is far lower (3 percent onaverage) than the number of pixels and the centroids of the substructures caneffectively capture the skeleton or saliency of the image; (2) all the imageshave the recursive levels more than three, indicating that they are indeedliving images; (3) no more than 2 percent of the substructures aredecomposable; (4) structural beauty can be measured by the recursively definedsubstructures, as well as their decomposable subsets. The recursive approach isproved to be more robust than the non-recursive approach. The recursiveapproach and the non-recursive approach both provide a powerful means to studythe livingness or vitality of space in cities and communities.

Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space