An improved block-based matching algorithm of copy-move forgery detection

Abstract

Copy-move forgery is a common way of image tampering. Matching algorithm is the key step in copy-move forgery detection. Usually, the classical block-based matching algorithm (CBMA) can’t find all matched sub-blocks. In this paper, we propose an improved block-based matching algorithm (IBMA) to solve the problem. Firstly, we put the sum of feature vectors in the first column to get a new matrix. Secondly, the matrix is sorted by first column. Finally, every row of the matrix will search the following rows until the difference in the first column is larger than the threshold value. Experiment results show that the improved block-based matching algorithm is better than the classical block-based matching algorithm when an image was distorted by Gaussian noise, salt-pepper noise, or JPEG compression. The reason is that improved block-based matching algorithm can look for all matched sub-blocks, which makes copy-move forgery detection methods more robust.

Appendix

We assume $Q1=\left [ {{I}^{1}},{{I}^{2}},{\ldots } ,{{I}^{n}} \right ]$ and $Q2=\left [ {{L}^{1}},{{L}^{2}},{\ldots } ,{{L}^{n}} \right ]$. V _{e u r} is a constant and more than zero.

Condition $\mathcal {A}$::

$\left | \left ({{I}^{1}}+{{I}^{2}}+{\ldots } +{{I}^{n}} \right )-\left ({{L}^{1}}+{{L}^{2}}+{\ldots } +{{L}^{n}} \right ) \right |>\sqrt {n}\times {{V}_{eur}}$.

Condition $\mathcal {B}$::

$SIM\left (Q1,Q2 \right )>{{V}_{eur}}$.

$$\begin{array}{@{}rcl@{}} &&\because \left| ({{I}^{1}}+{{I}^{2}}+...+{{I}^{n}})-({{L}^{1}}+{{L}^{2}}+...+{{L}^{n}}) \right|> \sqrt{n}\times V_{eur} \\ &&\therefore \frac{\left| ({{I}^{1}}-{{L}^{1}})+({{I}^{2}}-{{L}^{2}})+...+({{I}^{n}}-{{L}^{n}}) \right|}{\sqrt{n}}>V_{eur} \\ &&\therefore \frac{{{\left( ({{I}^{1}}-{{L}^{1}})+({{I}^{2}}-{{L}^{2}})+...+({{I}^{n}}-{{L}^{n}}) \right)}^{2}}}{n}>{{(V_{eur})}^{2}} \end{array} $$

From average inequality for every real number x _i ,i = 1, 2,...,n we can get: $x_{_{1}}^{2}+x_{_{2}}^{2}+...+x_{_{n}}^{2}\ge \frac {{{({{x}_{1}}+{{x}_{2}}+...+{{x}_{n}})}^{2}}}{n}$ we define: x _i = (I ⁱ − L ⁱ),i = 1, 2,...,n, obviously:

$$\begin{array}{@{}rcl@{}} &&{{({{I}^{1}}-{{L}^{1}})}^{2}}+{{({{I}^{2}}-{{L}^{2}})}^{2}}+...+{{({{I}^{n}}-{{L}^{n}})}^{2}}\ge \frac{{{\left( ({{I}^{1}}-{{L}^{1}})+({{I}^{2}}-{{L}^{2}})+...+({{I}^{n}}-{{L}^{n}}) \right)}^{2}}}{n} \\ &&\because \frac{{{\left( ({{I}^{1}}-{{L}^{1}})+({{I}^{2}}-{{L}^{2}})+...+({{I}^{n}}-{{L}^{n}}) \right)}^{2}}}{n}>{{(V_{eur})}^{2}} \\ &&\therefore {{({{I}^{1}}-{{L}^{1}})}^{2}}+{{({{I}^{2}}-{{L}^{2}})}^{2}}+...+{{({{I}^{n}}-{{L}^{n}})}^{2}}>{{(V_{eur})}^{2}}\\ &&\therefore\sqrt{{{({{I}^{1}}-{{L}^{1}})}^{2}}+{{({{I}^{2}}-{{L}^{2}})}^{2}}+...+{{({{I}^{n}}-{{L}^{n}})}^{2}}}>V_{eur} \end{array} $$