Elsevier

Signal Processing

Volume 91, Issue 10, October 2011, Pages 2290-2303
Signal Processing

Image analysis by Gaussian–Hermite moments

https://doi.org/10.1016/j.sigpro.2011.04.012Get rights and content

Abstract

Orthogonal moments are powerful tools in pattern recognition and image processing applications. In this paper, the Gaussian–Hermite moments based on a set of orthonormal weighted Hermite polynomials are extensively studied. The rotation and translation invariants of Gaussian–Hermite moments are derived algebraically. It is proved that the construction forms of geometric moment invariants are valid for building the Gaussian–Hermite moment invariants. The paper also discusses the computational aspects of Gaussian–Hermite moment, including the recurrence relation and symmetrical property. Just as the other orthogonal moments, an image can be easily reconstructed from its Gaussian–Hermite moments thanks to the orthogonality of the basis functions. Some reconstruction tests with binary and gray-level images (without and with noise) were performed and the obtained results show that the reconstruction quality from Gaussian–Hermite moments is better than that from known Legendre, discrete Tchebichef and Krawtchouk moments. This means Gaussian–Hermite moment has higher image representation ability. The peculiarity of image reconstruction algorithm from Gaussian–Hermite moments is also discussed in the paper. The paper offers an example of classification using Gaussian–Hermite moment invariants as pattern feature and the result demonstrates that Gaussian–Hermite moment invariants perform significantly better than Hu's moment invariants under both noise-free and noisy conditions.

Highlights

► A systematic study on Gaussian-Hermite moments. ► Discrete implementation and efficient computation of Gaussian-Hermite moments. ► A comparative study on image reconstruction. ► Derivation of eleven invariants of Gaussian-Hermite moments.

Introduction

Moments and functions of moments have been widely used in pattern recognition [1], [2], [3], edge detection [4], [5], image segmentation [6], texture analysis [7] and other domains of image analysis [8], [9] and computer vision [10], [11]. Among all kinds of moments, the geometric one is firstly introduced and has been used due to its simplicity and explicit geometric meaning. In the early 1980s of the last century, Teague introduced the orthogonal Legendre and Zernike moments using the corresponding Legendre polynomials and Zernike polynomials as kernel functions for image analysis [9]. Moreover, Zernike moments could store image information with minimal redundancy and have the rotation-invariant property. Since both Legendre and Zernike moments are defined on the continuous domain, the suitable transformations of image coordinates are needed when we calculate these moments in a discrete case. As we know, the computation of Legendre moments needs to transform the image coordinate into the interval [−1, 1] and Zernike polynomials are only valid inside the unit circle [12]. Besides, the discretization error derived from the approximation of integral is still inevitable during the implementation and the computation accuracy would be limited [13], [14]. Liao and Pawlak [13] conducted a theoretical analysis on the discretization error of the continuous moments and they proposed an approach to keep the approximation error under certain level according to the Simpson's rule. Other researches aiming at the improvement of computation accuracy have been accordingly focused on the geometric moment and Legendre moment. The exact computation proposed by Hosny provides an efficient and accurate computation of geometric and Legendre moments [15], [16]. The recent studies show that the computational complexity and the time of geometric moment can be significantly reduced using a symmetric kernel [17].

Meanwhile, the computational inconvenience of continuous moments impels the researches in the discrete orthogonal moments, which are proposed and gradually introduced to image analysis. Mukundan first introduced a set of moments to analyze the image based on the discrete Tchebichef polynomials [18]; some techniques for efficiently computing this kind of moments were also provided soon after [19]. Another widely used discrete orthogonal moment is Krawtchouk moment, which is based on the discrete classical Krawtchouk polynomials [20]. More recently, the discrete orthogonal Racah and dual Hahn moments were also introduced to image analysis [21], [22]. The computation of discrete orthogonal moments does not need any numerical approximations and image coordinate transformation, which make the discrete orthogonal moments superior, in general, to conventional continuous orthogonal moments in terms of image representation ability.

Although the discrete orthogonal moments have computational advantages and can be simply implemented, it does not mean that they can completely substitute the continuous ones. In fact, some continuous moments still outperform discrete moments in some aspects. As another kind of continuous orthogonal moments, the Gaussian–Hermite moments were firstly introduced by Shen [23]. The study of these moments is far from complete and mainly limited to their countable applications. Shen and Wu detected the moving objects using the moments [24], [25], [26]. Similarly, Wang and Dai [27] and Wang et al. [28]introduced the moments to the fingerprint classification in biometrics. Besides, other applications such as iris identification [29], SAR image segmentation [30] and stereo matching based on Gaussian–Hermite moments have recently been reported [31]. Although the above works are really impressive and applicable, they are almost based on using a few moments of low order as local convolution operators. As a kind of orthogonal moments, the abilities of global feature representation of Gaussian–Hermite moments have been little explored; to be more precise, there are almost no literatures reporting the image reconstruction from Gaussian–Hermite moments and its peculiarity in comparison with the other orthogonal moments. Moreover, there is a lack of research in the Gaussian–Hermite moment invariants, which should be an important tool for applications. Therefore, an overall study of Gaussian–Hermite moments has to be needed.

In this paper, we report a systematic and relatively complete study of Gaussian–Hermite moments. Their discrete implementation is formularized. Furthermore, an efficient image reconstruction algorithm from Gaussian–Hermite moments is presented and evaluated in comparison with the different kinds of moments. In addition, the corresponding moment invariants are also derived and they are evaluated by an object recognition task. According to our studies, despite being a kind of continuous moments, the Gaussian–Hermite moments outperform other ones even discrete orthogonal moments in several aspects.

The rest of the paper is organized as follows. In Section 2, after a brief review on the definitions of Hermite polynomials, weighted Hermite polynomials, Gaussian–Hermite polynomials and Gaussian–Hermite moments, we propose a set of Gaussian–Hermite rotation-invariants obtained on the same form as geometric moment invariants with demonstration and give a comparative study with Zernike and Fourier–Mellin moments. Section 3 focuses on the computation aspects of Gaussian–Hermite moments. It includes the discrete implementation and still contains the recurrence relation and symmetry properties that can be used to facilitate the computation of the moments. Section 4 provides the experimental validation to the theoretical framework presented in the previous sections. The reconstruction of different kinds of images from Gaussian–Hermite moments is detailed in this section. The last part of the section is an example of object recognition by Gaussian–Hermite moment invariants in comparison with Hu's moment invariants. Section 5 concludes this paper.

Section snippets

Gaussian–Hermite polynomials

The pth degree Hermite polynomial is defined asHp(x)=(1)pexp(x2)(dp/dxp)exp(x2)and it can be written in the form of series:Hp(x)=k=0p/2(1)kp!k!(p2k)!(2x)p2k.

Hermite polynomials satisfy the following orthogonality property with respect to the weight function w(x)=exp(x2):exp(x2)Hp(x)Hq(x)dx=2pp!πδpq,where δpq is the Kronecker delta. The recursive equation is available for fast computation of Hermite polynomials:Hp+1(x)=2xHp(x)2pHp1(x),forp1,with the initial conditions H0(x)=1 and H

Discrete implementation

Gaussian–Hermite moments are theoretically defined on the continuous domain (−, ). In order to compute the moments for a digital image I(i, j) whose size is K×K, the coordinate transformation over the square [−1≤x, y≤1] is recommended for a comparable evaluation of σ selection:x=(2iK+1)/(K1)andy=(2jK+1)/(K1).

Then the discrete version of Gaussian–Hermite polynomial H˜p(i,K;σ) is computed on the interval [−1, 1] and it in fact uses the equidistant sampling as a substitute for the continuous

Scale parameter σ selection for image reconstruction

It should be noted that the framework of image reconstruction from Gaussian–Hermite moments was initially and briefly introduced in 2007 [28]; this subject, however, still has much room to be deeply studied, especially to answer the following question: how to obtain the best reconstruction result automatically?

An important peculiarity of Gaussian–Hermite moments is that there is a scale parameter σ to be selected before moment computation and image reconstruction. Generally speaking, given the

Conclusion

In this paper, the orthogonal moments based on the Gaussian–Hermite polynomials are introduced and extensively studied. The discrete implementation that is related to the moment's computation and image reconstruction is detailed. Some properties that can facilitate the computation are also highlighted. Besides, the corresponding moment invariants that are independent of the transformations involving both rotation and translation are still proposed.

Based on the results of image reconstruction

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