Elsevier

Signal Processing

Volume 171, June 2020, 107509
Signal Processing

Fractional Charlier moments for image reconstruction and image watermarking

https://doi.org/10.1016/j.sigpro.2020.107509Get rights and content

Highlights

  • It is proposed a new set of fractional Charlier polynomials.

  • Constructing a new set of fractional moments for image reconstruction and image watermarking.

  • Providing numerical experiments to demonstrate their efficiency and superiority.

Abstract

In this paper, we propose a new set of discrete orthogonal polynomials called fractional Charlier polynomials (FrCPs). This new set will be used as a basic function to define the fractional discrete orthogonal Charlier moments (FrCMs). The proposed FrCPs are derived algebraically using the spectral decomposition of Charlier polynomials (CPs), then the Lagrange interpolation formula is used to derive the spectral projectors. Then, each spectral projector matrix is decomposed by the singular value decomposition (SVD) technique in order to build a basic set of orthonormal eigenvectors which help to develop FrCPs. FrCMs are deduced in matrix form from the proposed FrCPs and are applied for image reconstruction and watermarking. The experimental results show the capacity of the FrCMs proposed for image reconstruction and image watermarking against different attacks such as noise and geometric distortions.

Introduction

Moments have been widely used in image processing and analysis for many years because of their ability to extract characteristic information from the image locally and globally. They are applied with excellent results in different fields such as image reconstruction [1], [2], [3], [59], [60], image compression [3], [4], [5], image watermarking [6], [7], [8], [9], [10], edge detection [11], image geometric distortion correction [12] and image classification [13], [61].

The basic idea of moments is the projection of the data space on often orthogonal bases. Indeed, continuous orthogonal polynomials [14] such as Legendre [15], Zernike [16], Gegenbauer [17] and Fourier–Mellin [18] form continuous orthogonal moments (COMs), and discrete orthogonal polynomials [14], [19] such as Tchebichef [20], Krawtchouk [21], [22], [59], [62], Charlier [23], [24], [25], [59], [65], Hahn [23], [26], [63], [66] and Meixner [27], [64] constitute discrete orthogonal moments (DOMs).

Generally, the COMs and DOMs of an image are calculated for integer orders but sometimes they have to be calculated for real or fractional orders for reasons of precision and localization of the regions in the image. In this context, work has been performed with the objective of generalizing the calculation of Fourier transform (FT), discrete cosine transform (DCT), discrete sine transform (DST) for real or fractional orders by proposing new transformations such as the Fourier fractional transform (FrFT) [28], [29], discrete fractional cosine transform (DFrCT) [30] and the discrete fractional sine transform (DFrST) [30]. The same idea has been developed in recent years for some continuous orthogonal moments. B. Xiao et al. [31] developed the fractional Legendre polynomials. H. Zhang et al. [32] developed the fractional orthogonal Fourier–Mellin polynomials. K. Parand et al. [33] developed the fractional Chebyshev polynomials. The classical case of these polynomials is obtained if the fractional order is equal to the unit. These examples of fractional continuous orthogonal polynomials are used as basic functions to form new fractional continuous orthogonal moments of Legendre [31], Fourier–Mellin [32] and Chebyshev [34].

Fractional continuous orthogonal moments are defined in a Cartesian or polar continuous space, so their calculation requires a discretization of the continuous space and an appropriate approximation of the continuous integrals, which increases the complexity of the calculations and causes discretization errors [20], [22], [24], [26], [27], [31], [32], [34], [35] . To avoid these problems, X. Liu et al. [7] derived a new fractional Krawtchouk moments (also known as Fractional Krawtchouk Transform FrKT), where the kernel function is fractional Krawtchouk polynomials. To our attention, no other work concerning the other fractional discrete orthogonal moments has been published. With this in mind, we propose in this paper to calculate the fractional orthogonal moments of Charlier in order to apply them for reconstruction and watermarking image tasks.

The discrete orthogonal Charlier moments are calculated from the discrete orthogonal Charlier polynomials. H. Zhu et al. [24] proposed two general forms for calculating discrete orthogonal Charlier moments in order to apply them for reconstruction and compression image. M. Sayyouri et al. [36] proposed a set of separable Charlier–Hahn invariant moments based on the product of Charlier and Hahn discrete orthogonal polynomials, which are successfully used in pattern recognition and classification of 2D image, due to their robustness against geometric attacks such translation, scaling and rotation. The authors [37], [38] and [39] have extended these moments in the 3-D case and have applied them for 3D image reconstruction and classification.

In this paper, we have proposed a new set of discrete orthogonal polynomials called fractional Charlier polynomials (FrCPs), which are used as a basic function to define a new set of fractional Charlier moments (FrCMs). The FrCPs are reduced to the classical Charlier polynomials (CPs) when their fractional order is equal to the unit.

The proposed FrCPs are derived algebraically using first the spectral decomposition of the CPs, then the Lagrange interpolation formula is used to derive the projection matrices of CPs. Next, each projection matrix is decomposed by the singular value decomposition (SVD) technique in order to build a basic set of orthonormal eigenvectors help to develop FrCPs. The FrCPs are used to form new FrCs with additional parameters called fractional orders. Once the mathematical context of the new FrCMs has been developed, image reconstruction and watermarking image are chosen as test applications to verify the practical validity of these new moments. Image reconstruction by FrCMs is achieved through the orthogonality property of FrCPs. The proposed watermarking system consists of two essential phases: insertion and extraction of watermark. In the watermark insertion the fractional orders of the proposed FrCMs are used as security keys. The latter are necessary for the watermark extraction. In order to avoid the errors produced from the truncation of the Charlier polynomial matrix, during the calculation of FrCMs, the Gram-Schmidt process (GSP) is adopted to guarantee the orthogonality property of CPs for any order, which improves the image reconstruction capability by FrCMs. To improve the robustness of the proposed watermarking system to geometric attacks such as rotation and scaling, a technique was adopted to estimate rotational angles and scaling factors. Both FrCMs applications are tested on different images and compared with other recent methods.

The rest of the paper is structured as follows. In Section 2, we recall the definition of the classical discrete orthogonal Charlier moments. Then, we present the new fractional Charlier polynomials and their properties in Section 3. Next, we present the new 1-D and 2-D Fractional Charlier moments in Section 4. As for the Section 5, we introduce a watermarking scheme for 2D images based on the new FrCMs. The experimental results for evaluating the performance of the new FrCMs are given in Section 6, and Section 7 concludes the work.

Section snippets

Charlier moments

The discrete orthogonal moments of Charlier are projections of a function f (signal or image) onto the basis of Charlier polynomials (similarly, Fourier transformation is a projection onto a basis of the harmonic functions). In this section, we will present the mathematical background behind the Charlier moment theory, including polynomials, moments and image reconstruction.

Proposed Fractional Charlier polynomials

In this section, we study the eigenvalues and eigenvectors of CPs which help to develop the new Fractional Charlier Polynomials (FrCPs).

The Charlier polynomials are orthogonal over the interval [0, ∞[ according to Eq. (8). To compute the eigenvalues and eigenvectors of the Charlier polynomial matrix, one must work on a square matrix of finite order N which will affect the property of orthogonality of the Charlier polynomials. To solve this truncation problem, we propose to use the Gram–Schmidt

Proposed Fractional Charlier moments

The generalized Fractional Charlier Moments (FrCMs) are obtained from the FrCPs developed in the previous section.

Based on Eq. (13), the 1-D FrCMs of signal f(x) with fractional order a can be defined as follows:Ma=C^af

Based on Eq. (14) and property (a), the reconstruction of the signal f(x) can be found from its moments by using the following expression:f=C^aMa

In 2-D case, the FrCMs in terms of FrCPs with fractional order (a, b), for an image with intensity function f(x, y), can be defined as

Images watermarking system using fractional Charlier moments

Digital watermarking is a vast field that combines a large range of applications such as authentication, copyright protection, finger-print, copy control and broadcast monitoring [47].

The basic idea of digital watermarking is to incorporate some information's (called watermark or message) into another data file (Audio, image, video, 3D objects, etc.) called the host. Incorporation is done by imperceptible modifications on the host, so that the watermarked host can replace the original for

Simulation results

In this section we will provide an experimental validation of the theoretical framework presented in this paper. This section is divided into two sub-sections. In the first subsection, we will test the ability of the proposed FrCMs for the reconstruction of 2D images. In the second part, we will test the imperceptibility and the robustness of the proposed watermarking scheme against signal processing and geometric attacks. Several functions are used to qualify the proposed FrCMs:

The

Conclusion

In this article, we have proposed a new set of discrete orthogonal moments named fractional Charlier moments FrCMs. The spectral decomposition of the Charlier polynomials is adopted to determine the new fractional Charlier polynomials FrCPs which are used as kernel in FrCMs. The fractional orders of FrCMs give a wide choice in applications where FrCMs are used. The experimental results demonstrated that some better choice of these fractional orders can achieve best results in the fields of

Declaration of Competing Interest

The authors declare no conflict of interest.

Acknowledgement

This work has been partially supported by the Czech Science Foundation under the grant No. GA18-07247S.

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