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Triangular inequality-based rotation-invariant boundary image matching for smart devices

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Abstract

Nowadays there are many efforts to develop image matching applications exploiting a large number of images stored in smart devices such as smartphones, smart pads, and smart cameras. Boundary image matching converts boundary images to time-series and identifies similar boundary images using time-series matching on those time-series. In boundary image matching, computing the rotation-invariant distance between image time-series is a very time-consuming process since it requires a lot of Euclidean distance computations for all possible rotations. To support the boundary image matching in smart devices, we need to devise a simple but fast computation mechanism for rotation-invariant distances. For this purpose, in this paper we propose a novel rotation-invariant matching solution that significantly reduces the number of distance computations using the triangular inequality. To this end, we first present the notion of self-rotation distance and formally show that the self-rotation distance with the triangular inequality produces a tight lower bound and prunes many unnecessary distance computations. Using the self-rotation distance, we then propose a triangular inequality-based solution to rotation-invariant image matching. We next present the concept of k-self rotation distance as a generalized version of the self-rotation distance and formally show that this \(k\)-self rotation distance produces a tighter lower bound and prunes more unnecessary distance computations. Using the \(k\)-self rotation distance we also propose an advanced triangular inequality-based solution to rotation-invariant image matching. Experimental results show that our self-rotation distance-based algorithms significantly outperform the existing algorithms by up to one or two orders of magnitude, and we believe that this performance improvement makes our algorithms very suitable for smart devices.

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Notes

  1. Besides the range query of Definition 2, the \(k\)-nearest neighbor (\(k\)-NN) query is also widely used. However, we can process \(k\)-NN queries using range queries because we can regard the distances for current \(k\) candidates as the tolerances of range queries. Thus, in this paper we focus on the range query whose inputs are a query sequence and the tolerance.

  2. Previous solutions [19, 40] for rotation-invariant image matching focus on reducing the number of candidate data sequences through the filtering process. However, computing the rotation-invariant distances for these filtered candidates is still and essentially required. Thus, our solution is orthogonal to the previous solutions because our solution can be applied to their post-processing part of computing the rotation-invariant distances for candidate data sequences.

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Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012-0005258).

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Authors and Affiliations

  1. Department of Computer Science, Kangwon National University, 192-1 Hyoja2-Dong, Chunchon, Kangwon, 200-701, Republic of Korea

    Yang-Sae Moon

  2. Department of Software Design and Management, Gachon University, 1342 Seongnam-daero, Sujeong-gu, Seongnam-shi, Gyeonggi-do, 461-701, Republic of Korea

    Woong-Kee Loh

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Correspondence to Woong-Kee Loh.

Appendices

Appendix A: Proof of Theorem 3

Three sequences \(Q^j\), \(Q^{j+k}\), and \(S\) form a triangle in the \(n\)-dimensional space. By the triangular inequality, \(D(Q^{j+k},S) > |D(Q^j,S) - D(Q^j,Q^{j+k})|\) trivially holds. Thus, \(|D(Q^j,S)-D(Q^j,Q^{j+k})|\) is a lower bound of \(D(Q^{j+k},S).\) \(\square \)

Appendix B: Proof of Theorem 4

Two \(k\)-self rotation distances of \(Q^{j_1}\) and \(Q^{j_2}\) are \(D(Q^{j_1},Q^{j_1+k})\) and \(D(Q^{j_2},Q^{j_2+k})\), respectively. Here, \(D(Q^{j_1},Q^{j_1+k})\) can be converted into \(D(Q^{j_2},Q^{j_2+k})\) by the process of Eq. (3).

$$\begin{aligned}&D(Q^{j_1},Q^{j_1+k}) \nonumber \\&= \sqrt{\sum _{i=0}^{n-1} \left| q_{(i+j_1)\%n}-q_{(i+j_1+k)\%n} \right| ^2}\quad \text {(definition)} \nonumber \\&= \sqrt{\sum _{i=0}^{n-1} \left| q_{(i+(j_2-(j_2-j_1)))\%n}-q_{(i+(j_2-(j_2-j_1)+k))\%n} \right| ^2} \quad (j_1 = j_2-(j_2-j_1)) \nonumber \\&= \sqrt{\sum _{i=(j_2-j_1)}^{(j_2-j_1)+n-1} \left| q_{(i+(j_2-(j_2-j_1)))\%n}-q_{(i+(j_2-(j_2-j_1)+k))\%n} \right| ^2} \quad \text {(index \, shifting)} \nonumber \\&= \sqrt{\sum _{i=0}^{n-1}\left| q_{(i+j_2)\%n}-q_{(i+j_2+k)\%n} \right| ^2} \quad \text {(index \,shifting)}\nonumber \\&= D(Q^{j_2},Q^{j_2+k}). \end{aligned}$$
(3)

According to Eq. (3), \(D(Q^{j_1},Q^{j_1+k})\) and \(D(Q^{j_2},Q^{j_2+k})\) are the same, and this means that all possible \(k\)-self rotation distances are the same. \(\square \)

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Moon, YS., Loh, WK. Triangular inequality-based rotation-invariant boundary image matching for smart devices. Multimedia Systems 21, 15–28 (2015). https://doi.org/10.1007/s00530-014-0380-2

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