Mathematical Problems in Engineering

Volume 2018, Article ID 5463632, 19 pages

https://doi.org/10.1155/2018/5463632

## A Perceptive Approach to Digital Image Watermarking Using a Brightness Model and the Hermite Transform

^{1}Facultad de Ingeniería, Departamento de Procesamiento de Señales, Universidad Nacional Autónoma de México, Ciudad de México, Mexico^{2}Departamento de Ingeniería, Instituto Politécnico Nacional, UPIITA, Av. IPN No. 2580., Col. La Laguna Ticomán, 07340 Ciudad de México, Mexico

Correspondence should be addressed to Boris Escalante-Ramírez; xm.manu@sirob

Received 20 September 2017; Revised 10 January 2018; Accepted 28 January 2018; Published 29 April 2018

Academic Editor: Khaled Loukhaoukha

Copyright © 2018 Boris Escalante-Ramírez and S. L. Gomez-Coronel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents a watermarking technique in digital images using a brightness model and the Hermite Transform (HT). The HT is an image representation model that incorporates important properties of the Human Vision System (HVS), such as the analysis of local orientation, and the model of Gaussian derivatives of early vision. The proposed watermarking scheme is based on a perceptive model that takes advantage of the masking characteristics of the HVS, thus allowing the generation of a watermark that cannot be detected by a human observer. The mask is constructed using a brightness model that exploits the limited sensibility of the human visual system for noise detection in areas of high or low brightness. Experimental results show the imperceptibility of the watermark and the fact that the proposed algorithm is robust to most common processing attacks. For the case of geometric distortions, an image normalization stage is carried out prior to the watermarking.

#### 1. Introduction

The copyright protection of digital contents has become a great problem due to the increase of piracy, cloning of digital documents, and the espionage into the different mass media. The rapid growth of Internet, digital multimedia, and communication systems has increased exponentially the distribution of digital information (voice, data, images, and video), making evident the growing need to protect digital content. A solution for the protection of copyright and intellectual property is watermarking. For digital images, the process consists basically of embedding information into the code related to the author or copyright holder. The quality of an image watermarking technique is measured in terms of robustness, legibility, imperceptibility, and ambiguity [1, 2]. However, it is difficult to find a technique that embraces all of them, since robustness implies introducing stronger image distortions that compromise the watermark imperceptibility. For over a decade, different algorithms seeking to meet the above requirements have been proposed. Some of the proposed techniques are based on image transformations, using alternative representation models. They attempt to add the mark in the image transform domain [3]. Among these cases, we find the Discrete Fourier Transform (DFT), the Discrete Wavelet Transform (DWT), the Discrete Cosine Transform (DCT), the Contourlet transform, and other techniques. [4–12]. Nevertheless, their use does not guarantee by itself a robust watermarking technique. Some models use the HVS characteristics in order to obtain good results of imperceptibility and robustness, taking advantage of the sensitivity of frequency, luminance, and masking contrast. The watermark that exploits the perceptual information is named a perceptual watermark [9, 10, 13–15]. Transform domain techniques that use perceptual masks based on HVS properties have proved to be more robust since they resist geometric and filter attacks. For example, the algorithm described in [10] uses the DWT and a mask to determine the coefficients of detail where the watermark will be inserted. For the generation of the mask, the texture content and luminance in all frequency bands of the image are considered according to certain rules based on the HVS. The model by Barni et al. [10] actually considers the reduced eye sensitivity for detecting noise in the borders and the high and low brightness or luminance, as well as heavily texturized regions of an image. The result is a mask that takes into account the content of all the subbands of that image. Their results show that this technique is robust to common processing operations and has been used as a benchmark by several following investigations. In [16], similar ideas to those of scheme [10] have been used, the difference being that the latter works with the Hermite Transform (HT), and some modifications are set for the calculation of the perceptive mask. It can be regarded as the first work of watermarking using the HT. The present proposal is based on a perceptive approach that includes a brightness model and a normalization scheme of the image. Unlike most watermarking algorithms, which insert the mark considering the edges and homogeneous zones of the image, we use the brightness model to generate a perceptive mark and identify the image regions where the watermark detection becomes a difficult task for the human eye, that is, regions that are more likely to be modified without producing perceptive changes, thus assuring the invisibility of the mark. In order to generate the mask, the following elements are considered: luminance to brightness map, contrast sensitivity, and light adaptation threshold. These elements allow identifying the image structures and locations where additional information can be embedded without being perceived by a human observer. The main idea derives from the model proposed in [10] inspired by adaptive quantization in image compression schemes. In this approach, a perceptive mask is built based on the argument of the reduced visual sensitivity to noise in high resolution bands, in areas with high or low brightness, and in textured areas. Our approach shares the same principle; however we include a more elaborated luminance to brightness model that considers the multichannel mechanism that the human visual system uses to build the psychophysical perception of brightness [17]. In order to find textured areas, we use the high order Hermite Transform coefficients, which are known to represent perceptually relevant image structures; however, the visual perception of texture is also determined by the contrast and luminance; therefore we use the light adaptation threshold and contrast sensitivity measures that account for a better identification of textured areas where information can be perceptively hidden. In order to improve robustness against geometric attacks, we propose the use of image normalization techniques [1, 12, 18] that transform the original image so that the orientation and the scale of the objects are fixed. For this purpose, we employ geometric moments and invariants.

The Hermite Transform (HT) is a mathematical tool that allows the local analysis of the visual information of an image and builds scale space relations among its pixel intensities. Moreover, the HT incorporates the Gaussian derivative model of early vision [19–21] that considers the derivatives of Gaussian functions as suitable models for ganglionic and cortical visual cells. Similar to DWT, the HT also decomposes the image in a number of coefficients, where zero order coefficients represent a Gaussian-weighted image average. Larger order coefficients contain the image details; hence, the watermark data is to be inserted here. The mathematical theory of the HT is introduced in Section 2. The map of luminance brightness for the generation of the perceptive mask is described in Section 3, whereas, in Section 4, the proposed algorithm is detailed. In Section 5, the procedure for the image normalization based on [18] is presented. The results of experimental evaluations of the algorithm and the comparisons with current techniques appear in Section 6. Finally, the conclusions are detailed in Section 7.

#### 2. The Hermite Transform

The Hermite Transform [19–22] is a special case of polynomial transform, which is a technique of signal decomposition. The original signal , where are the coordinates of the pixels, can be located by multiplying the window function by the positions , that conform to the sampling lattice :

The periodic weighting function is then defined as

The unique condition that allows the polynomial transform to exist is that the weighting function must be different from zero for all coordinates .

The local information within every analysis window will then be expanded in terms of an orthogonal polynomial set. The polynomials , used to approximate the windowed information, are determined by the analysis window function and satisfy the orthogonal conditionfor ; ; and

The polynomial coefficients are calculated by convolving the original image with the filter function followed by a subsampling in the positions of the sampling lattice :

The orthogonal polynomials associated with are known as Hermite polynomials:where denotes the Hermite polynomial of order .

In the case of the Hermite Transform, it is possible to demonstrate that the filter functions correspond to Gaussian derivatives of order in and in , in agreement with the Gaussian derivative model of early vision [23]. Moreover, the window function resembles the receptive field profiles of human vision:

Besides constituting a good model for the overlapped receptive fields found in physiological experiments, the choice of a Gaussian window can be justified because it minimizes the uncertainty principle in the spatial and frequency domains. The recovery process of the original image consists in interpolating the transform coefficients through the proper synthesis filters. This process is known as inverse polynomial transform and is defined by

The synthesis filters of order in , and in , are defined by for and .

It is important to mention that the HT can generate coefficients with and without subsampling. In practice, HT implementation requires the choice of the size that Gaussian window spreads and a subsampling factor (if used) that defines the sampling lattice . The resultant Hermite coefficients are accommodated as a set of equal-sized subbands, as shown in Figure 1 (figure reproduced from S. Gomez-Coronel et al. (2016) [under the Creative Commons Attribution License/public domain]).