RefinerHash: a new hashing-based re-ranking technique for image retrieval

Abstract

Re-ranking is a task of refining an initially ranked list of images obtained from an image retrieval technique for a given query image, with the goal of enhancing retrieval performance in an efficient manner. However, existing re-ranking methods suffer from high computational complexity, leading to slow and resource-intensive operations that render them to be impractical for real-life applications. This necessitates the development of a computationally efficient re-ranking approach that effectively improves the retrieval performance. In this paper, we propose a novel and computationally efficient re-ranking method for image retrieval, utilizing the speedy and proficient nature of image hashing techniques for image representation. Use of the hash code of an image using its DCT and DWT coefficients constitutes the basis of the proposed re-ranking technique. Three balanced binary search trees, one using the hash codes of the images corresponding to the DCT coefficients, and the other two using the most significant and the least significant bits of the hash codes corresponding to the DWT coefficients, are formed. Each balanced binary search tree is searched by comparing the hash code of the query image with those of the images in the tree starting from its root to form a set of the images that are comparable to the query image. Finally, the three sets of the images resulting from the three balanced binary search trees are used to obtain the final re-ranked list of retrieved images. Experimental results on the proposed retrieval scheme based on hashing-based re-ranking and various other image retrieval schemes using benchmark datasets demonstrate the superiority of our approach in terms of computational efficiency and retrieval performance.

Appendix A

See Eqs. 18–25 here.

Proposition 1:

Given $M\ge K>0$, the limit of the ration of ${\varphi }_{2}$ and ${\varphi }_{1}$ as $M$ approaches infinity converges to zero, where ${\varphi }_{1}\in \mathcal{O}(M)$ and ${\varphi }_{2}\in \mathcal{O}\left({\text{log}}K\right).$

Proof

Let ${\varphi }_{1}={C}_{1}M$ and ${\varphi }_{2}={C}_{2}log K$, where ${C}_{1}$ and ${C}_{2}$ are positive constants representing the rate of growth of each function. Through analysis of the limit of the ratio of ${\varphi }_{1}$ and ${\varphi }_{2}$ as $M$ approaches infinity, we have

$$\underset{M\to \infty }{{\text{lim}}}\frac{{\varphi }_{2}}{{\varphi }_{1}} = \underset{M\to \infty }{{\text{lim}}}\frac{{C}_{2} {\text{log}}K}{{C}_{1}M}$$

(18)

Since ${\text{log}}K$ grows slower than $M$, as $M$ approaches infinity, the limit converges to zero:

$$\underset{M\to \infty }{{\text{lim}}}\frac{{C}_{2} {\text{log}}K}{{C}_{1}M}=0$$

(19)

$\therefore$ Asymptotically, the growth of ${\varphi }_{2}$ is negligible compared to the growth of ${\varphi }_{1}$. □

Proposition 2:

Time complexity of $\mathcal{O}\left(M\right)+\mathcal{O}\left({\text{log}}K\right)$ is $\mathcal{O}\left(M\right)$ for $M\ge K>0$.

Proof

Let function $\varphi$ be the sum of two other functions, ${\varphi }_{1}$ and ${\varphi }_{2}$, such that $\varphi ={\varphi }_{1}+{\varphi }_{2}$, where ${\varphi }_{1}\in \mathcal{O}\left(M\right)$ and ${\varphi }_{2}\in \mathcal{O}\left({\text{log}}K\right)$. Then we can express the upper bound of ${\varphi }_{1}$ and ${\varphi }_{2}$ as

$$ \begin{gathered} \exists C_{1} > 0,\exists N_{1} \in \mathbb{N},~\forall n > N_{1} ,\varphi _{1} \le C_{1} M \hfill \\ \exists C_{2} > 0,\exists N_{2} \in \mathbb{N},~\forall n > N_{2} ,\varphi _{2} \le C_{2} \log K. \hfill \\ \end{gathered} $$

(20)

We have $\varphi$ as

$$\varphi ={\varphi }_{1}+{\varphi }_{2}.$$

(21)

Then by applying the upper bounds for ${\varphi }_{1}$ and ${\varphi }_{2}$ from Eq. 20, we can derive an upper bound for $\varphi$ as

$$\varphi ={\varphi }_{1}+{\varphi }_{2}\le {C}_{1}M+{C}_{2}{\text{log}}K$$

(22)

or,

$$\varphi \le M\left({C}_{1}+{C}_{2}\frac{{\text{log}}K}{M}\right).$$

(23)

Now, based on Proposition 1, as $M$ grows large, the term ${C}_{ 2}$ becomes negligible compared to ${C}_{1}M$. Therefore, the dominant term here is ${C}_{1}M$, leading us to express the upper bound of φ as

$${C}_{1}+{C}_{2}\frac{{\text{log}}K}{M}{\le C}_{1}$$

(24)

or

$$\varphi \le {C}_{1}M.$$

(25)

So, the time complexity $\mathcal{O}(M)+\mathcal{O}(log K)$ is dominated by $\mathcal{O}({\text{M}})$, which leads us to conclude that the overall time complexity is $\mathcal{O}({\text{M}})$. □