Temporal proximities: Self-similar temporally close shapes

Abstract

This article introduces temporal proximity spaces as a framework to observe surface shapes as well as geometric shapes that change over time. A surface shape has a boundary and a non-empty interior, which is approximated by a geometric shape that lies within the boundary of a surface shape recorded in a video frame. Temporally close shapes (briefly, delta(Delta t), shapes) persist over the some temporal interval. Those surface shapes that appear withing the same video frame are strongly self-similar as well as temporally close. The rate of change of self-similar shapes shE, shE' is represented by div(shE), div(shE') pairs. Temporally close shapes that appear during the same temporal interval may or may not be spatially or descriptively close to each other. Persistent as well as spatially close shapes share belong to the same era and also overlap along their boundaries or withing their interiors. Overlapping appearances of shapes such as vortexes occur within temporal CW (Closure-finite Weak) spaces (briefly, tCW spaces), which are an extension of the CW spaces introduced by J.H.C. Whitehead during the 1940s. Because of their simplicity, tCW space provide a workable setting for the study of delta(Delta t), surface as well as geometric shapes in sequences of video frames. (C) 2021 Elsevier Ltd. All rights reserved.