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Optimizing Image Retrieval with an Extended b-Metric Space

Abdelkader Belhenniche, Roman Chertovskih

TL;DR

This work introduces ${\rm NEM}_{\sigma}$, a dynamic distance measure within extended $b$-metric spaces to enhance image retrieval in QBIC systems. By replacing the fixed stretching in ${\rm NEM}_{r}$ with a velocity-/environment-aware function $\sigma$ and a modular factor $\theta$, the authors derive a generalized relaxed triangle inequality that sustains robust pattern matching in complex, dynamic contexts. They define ${\rm NEM}_{\sigma}$ via a minimal $(m,n)$-\sigma-mapping that combines a stretch cost $\mathcal{S}$ and a distance cost $\mathcal{D}$, both expressible as double integrals, and prove it forms an extended $b$-metric with $\theta(x,z)=1+\Delta(x,z)$. The framework enables finer, adaptive alignment of sequences under varying conditions, demonstrated conceptually through $\delta$-edges and velocity-aware mappings, with the potential to significantly improve large-scale image retrieval in real-world, dynamic datasets.

Abstract

This article provides a new approach on how to enhance data storage and retrieval in the Query By Image Content Systems (QBIC) by introducing the ${\rm NEM}_σ$ distance measure, satisfying the relaxed triangle inequality. By leveraging the concept of extended $b$-metric spaces, we address complex distance relationships, thereby improving the accuracy and efficiency of image database management. The use of ${\rm NEM}_σ$ facilitates better scalability and accuracy in large-scale image retrieval systems, optimizing both the storage and retrieval processes. The proposed method represents a significant advancement over traditional distance measures, offering enhanced flexibility and precision in the context of image content-based querying. Additionally, we take inspiration from ice flow models using ${\rm NEM}_σ$ and ${\rm NEM}_r$, adding dynamic and location-based factors to better capture details in images.

Optimizing Image Retrieval with an Extended b-Metric Space

TL;DR

This work introduces , a dynamic distance measure within extended -metric spaces to enhance image retrieval in QBIC systems. By replacing the fixed stretching in with a velocity-/environment-aware function and a modular factor , the authors derive a generalized relaxed triangle inequality that sustains robust pattern matching in complex, dynamic contexts. They define via a minimal -\sigma-mapping that combines a stretch cost and a distance cost , both expressible as double integrals, and prove it forms an extended -metric with . The framework enables finer, adaptive alignment of sequences under varying conditions, demonstrated conceptually through -edges and velocity-aware mappings, with the potential to significantly improve large-scale image retrieval in real-world, dynamic datasets.

Abstract

This article provides a new approach on how to enhance data storage and retrieval in the Query By Image Content Systems (QBIC) by introducing the distance measure, satisfying the relaxed triangle inequality. By leveraging the concept of extended -metric spaces, we address complex distance relationships, thereby improving the accuracy and efficiency of image database management. The use of facilitates better scalability and accuracy in large-scale image retrieval systems, optimizing both the storage and retrieval processes. The proposed method represents a significant advancement over traditional distance measures, offering enhanced flexibility and precision in the context of image content-based querying. Additionally, we take inspiration from ice flow models using and , adding dynamic and location-based factors to better capture details in images.

Paper Structure

This paper contains 4 sections, 1 theorem, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary Concepts and Results
  3. Main results
  4. Conclusion

Key Result

Theorem 3.1

Let $X$ be nonempty sequence and let the function $\mathcal{B} \colon X \times X \rightarrow [0, +\infty)$ be an extended $b$-metric with modulus $\alpha \colon X \times X \rightarrow [1,+\infty)$. Then, for all $X, Y, Z$, the measure ${\rm NEM}_{\sigma}$ is extended b-metric.

Figures (6)

  • Figure 1: Pattern matching with relaxed triangle inequality. (a) represents the three robots in a stable, stationary state, while in (b) and (c) the robots are in motion.
  • Figure 2: A minimal $(9,9)$-mapping with the stretch-cost of $6r$ (reproduced from FAG98, Fig. 1).
  • Figure 3: Comparison of ${\rm NEM}$ and ${\rm NEM}_r$: The left plot illustrates the alignment of a circle and an ellipse in ${\rm NEM}$, where the circle remains unchanged. In contrast, the right plot demonstrates ${\rm NEM}_r$, where a stretching penalty ($r = 1$) elongates the circle along the $y$-axis, modifying its shape to better match the dimensions of the ellipse.
  • Figure 4: Effect of parameter $r$ on stretching penalty in shape matching
  • Figure 5: Minimal (9,9)-$\sigma$-mapping.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Example 1: Roshan:13
  • Definition 2.2
  • Remark 2.3
  • Example 2
  • Example 3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1