Envelope-based boundary image matching for smart devices under arbitrary rotations

Recently, a variety of smart devices have been introduced for entertainment or industrial purposes, and there have been a lot of needs for image matching applications exploiting a large number of images stored in those smart devices. Boundary image matching identifies similar boundary images using their corresponding time-series, and supporting the rotation invariance is crucial to provide more intuitive matching results not only in conventional computing devices but also in smart devices such as smartphones, smart pads, and smart cameras. Computing the rotation-invariant distance between image time-series, however, is a very time-consuming process since it requires a lot of Euclidean distance computations for all possible rotations. We here note that, for smart devices, a very efficient mechanism of computing rotation-invariant distances is required. For this purpose, in this paper we use a novel notion of envelope-based lower bound proposed by Keogh et al. (VLDB J 18:611–630, 2009 ) to reduce the number of distance computations dramatically. With the help of Keogh et al.’s prior work (Keogh in Proceedings of the 28th International Conference on Very Large Data Bases, 406–417, 2002 ; Keogh et al. in VLDB J 18:611–630, 2009 ), we first explain how to construct a single envelope from a query sequence and how to obtain a lower bound of the rotation-invariant distance using the envelope. We then explain that the single envelope lower bound can reduce a number of distance computations. This single envelope approach, however, may cause bad performance since it may incur a larger lower bound due to considering all possible rotated sequences in a single envelope. To solve this problem, we present a concept of rotation interval, and using the concept of multiple envelopes proposed by Keogh et al. (VLDB J 18:611–630, 2009 ) with these rotation intervals, we then generalize the envelope-based lower bound by exploiting multiple envelopes rather than a single envelope. We also propose equi-width and envelope-minimization divisions as the method of determining rotation intervals in the multi-envelope approach. We further present an advanced multi-step matching algorithm that progressively prunes search spaces by dividing the rotation interval in half. Experimental results show that our envelope-based solutions outperform naive solutions by one to three orders of magnitude. We believe that this performance improvement makes our algorithms very suitable for smart devices.