Mathematical Problems in Engineering

Volume 2018, Article ID 1673283, 17 pages

https://doi.org/10.1155/2018/1673283

## A Comparative Study on Discrete Shmaliy Moments and Their Texture-Based Applications

^{1}Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico^{2}Luxembourg Institute of Science and Technology (LIST), Belvaux, Luxembourg

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

Received 31 March 2018; Accepted 8 July 2018; Published 9 August 2018

Academic Editor: Vincenzo Vespri

Copyright © 2018 Germán González et al. 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

In recent years, discrete orthogonal moments have attracted the attention of the scientific community because they are a suitable tool for feature extraction. However, the numerical instability that arises because of the computation of high-order moments is the main drawback that limits their wider application. In this article, we propose an image classification method that avoids numerical errors based on discrete Shmaliy moments, which are a new family of moments derived from Shmaliy polynomials. Shmaliy polynomials have two important characteristics: one-parameter definition that implies a simpler definition than popular polynomial bases such as Krawtchouk, Hahn, and Racah; a linear weight function that eases the computation of the polynomial coefficients. We use IICBU-2008 database to validate our proposal and include Tchebichef and Krawtchouk moments for comparison purposes. The experiments are carried out through* 5*-fold cross-validation, and the results are computed using random forest, support vector machines, naïve Bayes, and k-nearest neighbors classifiers.

#### 1. Introduction

Over the past few years, discrete orthogonal moments (DOMs) have attracted the attention of the image analysis community because they possess the attribute of describing local and global features in images efficiently. The DOMs are optimal in the sense that they represent images with minimal information redundancy. The characteristic is originated by the orthogonality condition of the polynomial basis where the moments are computed [1]. Furthermore, contrary to the continuous moments, the DOMs are defined in the discrete domain. Hence, their computation does not require spatial quantization [2].

Families of DOMs such as Racah [3], dual Hahn [4], and Tchebichef [5] have been used successfully in many applications, for instance, feature analysis [6, 7], face recognition [8, 9], and image retrieval [10, 11] to name a few. Particularly, medical imaging has exploited orthogonal moments to characterize different types of biological tissue and assess the severity of several diseases [12–16].

An important impulse that has placed them in the spotlight is the discussion of the numerical instability problem that occurs in high-order moments. Since the DOMs are calculated as the projection of the image on a weighted kernel or set of orthogonal polynomials, their computation is usually linked to the size of the image. The higher the order of the moment, the higher the numerical error.

Traditionally, the computation of the DOMs is carried out by the use of recursive equations. However, the methodology tends to accumulate and propagate errors that degrade the representation of the image. Other methodologies have tackled the issue. In [17], Mukundan presented a seminal paper where the symmetry and renormalization of the Tchebichef polynomials are used to reduce the accumulation of numerical errors; the discrete Krawtchouk moments [18] are defined on a set of weighted Krawtchouk polynomials to improve the numerical stability; and Bayraktar et al. [19] proposed computing an offline lookup table of Tchebichef and Krawtchouk polynomial coefficients to overcome calculation errors.

In addition to the aforementioned normalization methods, other approaches such as partitioning and sliding window have been also proposed. The former divides the image into smaller nonoverlapped sub-images where low-order moments are calculated independently, while the latter uses a rectangular window that slides usually from left to right and from top to down. On every window region low-order DOMs are computed.

A new class of discrete orthogonal moments derived from Shmaliy polynomials [20] was recently proposed. Shakibaei Asli and Flusser called this new class of moments “discrete Shmaliy moments” (DSMs) [21]. We highlight that the Shmaliy polynomial basis has two important characteristics: one-parameter definition that implies a simpler definition than popular polynomial bases such as Krawtchouk, Hahn, Dual Hahn, and Racah; a linear weight function that eases the computation of the polynomial coefficients. The Shmaliy polynomials are opposite to the Tchebichef polynomials, which use a symmetric nonlinear weight function.

So far, the experiments conducted by Shakibaei Asli and Flusser have shown the capability of the DSMs as feature descriptors in one dimension. In this paper, we explore the use of the discrete Shmaliy moments as 2D texture descriptors and present an extensive comparative study using the IICBU-2008 database [22]. A quantitative analysis based on N-way ANOVA is also conducted to support our proposal.

In Section 2, the mathematical theory of DOMs is presented. We introduce DSMs as the main part of this paper. Tchebichef and Krawtchouk moments are also briefly described for comparison purposes. The rest of this paper is organized as follows: in Section 3, we review the texture model and the general classification scheme; the image database is described in Section 4; and in Section 5, the experimental results are summarized. Finally, Section 6 contains concluding statements.

#### 2. Discrete Shmaliy Moments

In 1980, Teague defined the orthogonal moments in the continuous space [23]. Since then, many variations have been published and used successfully. For example, Zernike and Legendre moments have been used as detectors of invariant features [24–26]. However, the implementation of a continuous basis involves a discrete approximation that affects the properties of invariance and orthogonality. In addition, in some cases, the coordinated space of the image must be transformed in order to compute the continuous moments [27, 28].

Generally speaking, the orthogonal moments are a set of scalar quantities that are not correlated among them. They are useful for characterizing local and global features in images [29]. The orthogonal moments are computed as the projection of the image on an orthogonal polynomial basis . Formally, they are defined as follows: . One way of interpreting the projection is as the correlation measure between the image and the polynomial basis [15]. On the other hand, the discrete orthogonal moments overcome the aforementioned limitations because they still satisfy the property of orthogonality and are also defined in the same image domain.

Discrete Shmaliy polynomials were developed as unbiased finite impulse responses for predictive FIR filters [30] and have been applied on blind fitting in finite-length data [20]. The discrete Shmaliy polynomials have two important characteristics. Their definition is simple because it requires only one parameter that represents the length of the data, and they use a linear weight function, which eases the computation. In addition, the discrete Shmaliy polynomials represent the discrete version of the radial Mellin polynomials when the length of the signal approaches infinity [31].

The* p*th-order Shmaliy polynomial is defined as follows: where is the length of the data, is the Pochhammer symbol with , , and the hypergeometric function is given by

In practice, the weighted discrete Shmaliy polynomials [21] are used to minimize the instability at high-order moments. The orthogonal basis is built as follows: where is the weight function and defines the norm.

The weighted discrete Shmaliy polynomials satisfy the following orthogonality condition:

According to Shakibaei Asli and Flusser, it is possible to define multivariate polynomials (2D and 3D) as products of the univariate polynomials (1D) in each dimension.

Therefore, using (3), the discrete Shmaliy moments of order of the image are defined as where and are the size of the image on - and -axes, respectively. 1D and 2D discrete Shmaliy polynomials are shown in Figures 1(a) and 1(b), respectively.