Iterated local search optimized hashing for image copy detection

Abstract

Currently, researches on content based image copy detection mainly focus on robust feature extraction. However, due to the exponential growth of online images, it is necessary to consider searching among large number of images, which is very time-consuming and unscalable. Hence, we need to pay much attention to the efficiency of image detection. Although many hashing methods has been proposed, they did not show excellent performance in decreasing semantic loss during the process of hashing. In this paper, we propose a hashing based method for image copy detection, which not only generates compact fingerprint for image representation, but also prevents huge semantic loss during the process of hashing. To generate the fingerprint, an objective function of semantic loss is constructed and minimized, which combine the influence of both the neighborhood structure of feature data and mapping error. To minimize the objective function, we first calculate an approximate solution through trace optimization, and then optimize the solution through Iterated Local Search(ILS) to further decrease semantic loss. Experimental results show that our approach significantly outperforms state-of-art methods.

Appendix

Using the obtained results of (5) and (7), we get

$$\begin{array}{@{}rcl@{}} && {X^{T}}W+ \bold 1 b-H\\ && ={X^{T}}W+ \bold 1 {{\frac{1}{n}}({\bold 1^{T}}H-{\bold 1^{T}}{X^{T}}W)}-H\\ && =A{X^{T}}W-AH\\ && =A{X^{T}}({XA{X^{T}}+\gamma I}^{-1})XAH-AH\\ && =(A{X^{T}}({XA{X^{T}}+\gamma I}^{-1})XA-A)H \end{array} $$

(13)

Let B = (X A X ^T + γ I)⁻¹, it is obvious that B = B ^T, so W = B X A H and X ^T W + 1 b − H = (A X ^T B X A − A)H,then we transform the last part of the objective function as follows by applying the theory of trace optimization [10]:

$$\begin{array}{@{}rcl@{}} &&{\left\| {X^{T}}W + {\bold 1}b-H \right\|_{F}^{2}}+\gamma \left\| W \right\|_{F}^{2}\\ && = tr{H^{T}}{(A{X^{T}}BXA-A)^{T}}(A{X^{T}}BXA-A)H+\\ && \quad \gamma tr({W^{T}}W)\\ && = tr{H^{T}}(A{X^{T}}BXAA{X^{T}}BXA-2A{X^{T}}BXA+A)H+\\ && \quad \gamma tr({H^{T}}A{X^{T}}BBXAH)\\ && = tr{H^{T}}(\gamma A{X^{T}}BBXA+A{X^{T}}BXA{X^{T}}BXA\\ && \quad \quad\quad\quad-2A{X^{T}}BXA+A)H\\ && = tr{H^{T}}(A{X^{T}}B(\gamma I+ XA{X^{T}})BXA\\ && \quad \quad\quad\quad-2A{X^{T}}BXA+A)H\\ && = tr{H^{T}}(A{X^{T}}B{B^{-1}}BXA-2A{X^{T}}BXA+A)H\\ && = tr{H^{T}}(A{X^{T}}BXA-2A{X^{T}}BXA+A)H\\ && = tr{H^{T}}(A-A{X^{T}}BXA)H\\ && = tr{H^{T}}CH \end{array} $$

(14)

Where C = A − A X ^T B X A.