On the mathematical approach to the ``photo-likeness" of images

Abstract

In image recognition, it is significant to determine the boundary between meaningful and non-meaning images. In this paper, we show a mathematical approach to this problem by defining a ``quasi-photographic" image. In order to formulate the question `What is photograph likeliness ? ' mathematically, we introduce a function `depth' that takes real values for images and analyze its asymptotic behavior. We also examine that an actual photograph is indeed a quasi-photograph. The idea of depth comes from the rank of the 0th persistent homology of a cubical complex and it can be expected that more precise classification of images can be obtained by analyzing the higher rank in the future. We also believe that it can be applied to deep learning, which is being actively utilized recently in image recognition, to selection of learning data. We would like to propose one approach of applicating pure mathematics in image recognition.