Fractional-order quaternion exponential moments for color images

https://doi.org/10.1016/j.amc.2021.126061Get rights and content

Highlights

  • The properties of fractional-order and integer-order radial basis functions are analyzed and compared in depth.

  • FrEMs are constructed on the basis of fractional-order radial basis functions.

  • FrQEMs are constructed based on the quaternion theory.

  • The experimental results show that FrQEMs have great performance.

  • A novel kind of fractional-order radial basis functions is constructed.

Abstract

Due to their strong image description ability, in recent years, quaternion exponential moments (QEMs) have attracted wide attention from relevant researchers. However, there is a numerical instability problem with QEMs because they can only take integer orders, and this problem restricts the performance of QEMs in image reconstruction and noise resistance. In this paper, the concept of fractional order is introduced and incorporated into exponential moments (EMs) and fractional-order exponential moments (FrEMs) are proposed; then the FrEMs are extended to fractional-order quaternion exponential moments (FrQEMs) that are suitable for color images and have better performance than QEMs in noise resistance and image reconstruction; subsequently, it is proved that FrQEMs are invariant to rotation; and finally, the properties of FrQEMs are experimentally analyzed and FrQEMs are used for object recognition. The experimental results show that FrQEMs have very excellent performance in color image reconstruction, high noise resistance, good rotation invariance and object recognition in color images.

Introduction

Image moments are a kind of highly concentrated, geometrically invariant image features with strong ability to describe the global features of images, and have been widely used in various fields of image processing, such as image watermarking, pattern recognition, object classification, object recognition and forgery detection. Image moments were first studied in the 1960s. Hu first proposed Hu moment invariants [1] in 1961, attracting wide attention to image moments from researchers. Later rotational moments (RMs) [2] and complex moments (CMs) [3] were proposed successively, making it possible to construct rotation, scale and translation invariants. However, as Hu moments, rotational moments and complex moments are non-orthogonal moments, there is an information redundancy problem with them and it is very difficult to use them to reconstruct the original images.

To solve this difficult problem with respect to image reconstruction, the concept of orthogonal moments has been proposed by scholars based on the theory of orthogonal functions. Orthogonal moments are free from the problem of information redundancy, therefore, the original images can be reconstructed easily using a small number of such moments. Orthogonal moments include discrete and continuous orthogonal moments. Discrete orthogonal moments are free of discrete errors, and their calculation accuracy is only related to the high-order numerical transmission error. Discrete orthogonal moments mainly include discrete Chebyshev-Fourier moments (DCHFMs) [4], discrete Krawtchouk moments (DKMs) [5], discrete Chebyshev moments (DCHMs) [6] and discrete Hahn moments (DHMs) [7]. However, since discrete orthogonal moments themselves are not multi-distortion invariants, they should be expressed as linear combinations of geometric moments for the purpose of constructing geometric invariance. This process requires a higher computational complexity than the continuous orthogonal moments calculation. Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation, scale and translation. Continuous orthogonal moments mainly include Legendre moments (LMs) [8], Zernike moments (ZMs) [8], Pseudo-Zernike moments (PZMs) [9], orthogonal Fourier-Mellin moments (OFMMs) [10], Chebyshev-Fourier moments (CHFMs) [11], radial harmonic Fourier moments (RHFMs) [12], Bessel-Fourier moments (BFMs) [13], polar harmonic transforms (PHTs) [14] and exponential moments (EMs) [15].

Traditional image moments can only take integer numbers, which restricts their image reconstruction performance and noise resistance. In recent years, fractional-order problems have attracted wide attention, and more and more scholars have started to study fractional-order moments. Xiao et al. defined fractional-order Legendre-Fourier moments (FrOLFMs) [16]. Zhang et al. defined the fractional-order Fourier-Mellin polynomial and then derived orthogonal fractional-order Fourier-Mellin moments (FrOFMMs) [17]. Yang et al. proposed fractional-order Zernike moments (FrZMs) [18] and Hosny et al. proposed fractional-order polar harmonic transforms (FrPHTs) [19], respectively. Among the aforementioned continuous orthogonal moments, EMs are characterized by high noise resistance, low information redundancy, simple form of basis function, low computational complexity, and strong image description ability. In this paper, the concept of fractional-order moments is incorporated into EMs to improve their performance in image reconstruction and noise resistance. EMs are extended to fractional-order exponential moments (FrEMs) mainly by modifying their radial basis functions. In this study, we add a variable fractional parameter t(t > 0) to the radial basis function of EMs, fractional parameter t can be used to adjust the zero distribution and change rate of radial basis function. Then in order to keep the orthogonality of FrEMs, we further adjust the radial basis function of FrEMs [20]. FrEMs have better performance than EMs in noise resistance and image reconstruction, and are geometrically invariant. In addition, when they are used for object recognition, FrEMs can achieve a recognition rate higher than that achieved by EMs, and FrEMs are more robust to noise than EMs in object recognition.

Although image moments theory has been widely studied in recent years, it mainly focuses on gray images [21]. However, color images provide more information than gray images, so they are more popular in many applications [22]. However, traditional color image processing methods are mainly based on RGB decomposition or gray-scaling, which ignore the relations between the RGB components of color images, affecting the accuracy of color image processing. Image representation and application based on quaternions are becoming more and more important, and some quaternion-based methods have been applied to continuous orthogonal moments. The existing quaternion continuous orthogonal moments mainly include quaternion orthogonal Fourier-Mellin moments (QOFMMs) [23], quaternion Zernike moments (QZMs) [24], quaternion Bessel-Fourier moments (QBFMs) [25], quaternion pseudo-Zernike moments (QPZMs) [26], quaternion radial Tchebichef moments (QRTMs) [26], quaternion Legendre-Fourier moments (QLFMs) [27], quaternion radial harmonic Fourier Moments (QRHFMs) [28] and quaternion polar harmonic Fourier Moments (QPHFMs) [29]. In this paper, fractional-order quaternion exponential moments (FrQEMs) are constructed based on the quaternion theory and FrEMs. FrQEMs treats a color pixel as a vector, which preserves the relationship between different channels of color images [30]. Therefore FrQEMs have strong color image description ability, excellent robustness and geometric invariance [31]. The experimental results show that compared with integer-order quaternion exponential moments (QEMs) and other fractional-order continuous orthogonal moments, FrQEMs have better image reconstruction performance and object recognition performance.

The innovative points of this study are described below.

  • (1)

    Fractional-order radial basis functions are constructed, and the properties of fractional-order and integer-order radial basis functions are analyzed and compared in depth.

  • (2)

    FrEMs are constructed on the basis of fractional-order radial basis functions, and FrQEMs suitable for color images are constructed based on the quaternion theory. These geometrically invariant FrQEMs can be used as novel descriptors for invariant color image recognition.

  • (3)

    The experimental results show that FrQEMs have better performance than integer-order QEMs and other fractional-order continuous orthogonal moments in color image reconstruction and object recognition in color images.

Other sections of this paper are organized as follows: Section 2 describes FrQEMs construction process in detail and analyzes the calculation process and the reconstruction process of FrQEMs, Section 3 mainly analyzes the properties of FrQEMs from the perspectives of the number of zeros and the rate of change of radial basis functions, Section 4 provides the detailed experimental results and discussions with respect to image reconstruction, noise resistance, rotation invariance and object recognition, and Section 5 draws a conclusion of this study.

Section snippets

Proposed fractional-order quaternion exponential moments

EMs are a kind of multi-distortion invariant image features. The shape features of an object can be well represented with a group of EMs feature vectors. In this section, in order to introduce FrQEMs, the definition of EMs will be given firstly, then EMs will be extended to the fractional order to obtain FrEMs, subsequently, the quaternion theory will be introduced to extend FrEMs to FrQEMs, and finally, the properties of FrQEMs will be discussed.

Analysis of radial basis functions

The differences in the properties of orthogonal moments are mainly reflected in the radial basis functions of such moments. In this section, FrQEMs will be compared with QEMs, FrQOFMMs, FrQRHFMs and FrZMs with respect to the properties of their radial basis functions.

Experiments and analysis of experimental results

In this section, the performance of FrQEMs is tested and compared with that of QEMs, FrQRHFMs and FrQOFMMs from the perspectives of image reconstruction, noise resistance, rotation invariance and object recognition. Experiments are conducted on 30 color images with size of 256 × 256, including the 12 experimental images shown in Fig. 6.

Conclusion

QEMs have strong image description ability and the color images processed with QEMs are more complete than those processed by traditional methods that are based on channel decomposition and gray-scaling. In this study, QEMs that can only take integer values are extended to FrQEMs with better performance by modifying the radial basis functions of QEMs while maintaining certain properties of QEMs, such as orthogonal invariance. Such extension improves the performance of QEMs in noise resistance

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Nos: 61802212, 61872203, 61806105, 61701212, 61701070 and 61672124), the Shandong Provincial Natural Science Foundation (No: ZR2019BF017), the Project of Shandong Province Higher Educational Science and Technology Program (No: J18KA331), Major Scientific and Technological Innovation Projects of Shandong Province (Nos: 2019JZZY010127, 2019JZZY010132 and 2019JZZY010201), Plan of Youth Innovation Team Development of

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      The corresponding NIRE values are displayed with the reconstructed images to ensure accuracy. Figs. (4) and (5) show that the FrEMs [41] and FrZMs [42] are not able to reconstruct color images with high quality. Fig. (3b) show that for low-order moments, max ≤ 15, the proposed FrQSGMs outperformed the recent fractional-order moments, FrQPHTs [43], FrMLFMs [44], FrMJFMs [45], FrRHFMs [46], and Fr-QGLMs [47].

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