Novel moment invariants for improved classification performance in computer vision applications

doi:10.1016/j.patcog.2009.05.008

Pattern Recognition

Volume 43, Issue 1, January 2010, Pages 58-68

https://doi.org/10.1016/j.patcog.2009.05.008 Get rights and content

Abstract

A novel set of moment invariants based on the Krawtchouk moments are introduced in this paper. These moment invariants are computed over a finite number of image intensity slices, extracted by applying an innovative image representation scheme, the image slice representation (ISR) method. Based on this technique an image is decomposed to a several non-overlapped intensity slices, which can be considered as binary slices of certain intensity. This image representation gives the advantage to accelerate the computation of image's moments since the image can be described in a number of homogenous rectangular blocks, which permits the simplification of the computation formulas. The moments computed over the extracted slices seem to be more efficient than the corresponding moments of the same order that describe the whole image, in recognizing the pattern under processing. The proposed moment invariants are exhaustively tested in several well known computer vision datasets, regarding their rotation, scaling and translation (RST) invariant recognition performance, by resulting to remarkable outcomes.

Introduction

Image moments have attracted the attention of the engineering scientific community for several decades, as a powerful tool to describe the content of an image. They have been used in many research fields of the engineering life, such as image processing [1], [2], [3], [4], pattern recognition and machine vision [5], [6], with considerable results. The first introduction of image moments for classification purposes was performed by Hu [7], by introducing the concept of moment invariants for invariant pattern recognition applications. By using the geometrical, central and normalized image moments, Hu [1] proposed seven measurements invariant to any translation, scaling and rotation transformation of the image be processed, called moment invariants. These moment invariants are successfully used as image descriptors in many pattern recognition applications [8], [9].

Since the first introduction of moment invariants in classification problems, many attempts to develop improved moment invariants with superior classification performance have occurred lately [10], [11], [12], [13]. However, these types of moment invariants suffer from high information redundancy, by making them less efficient in difficult problems where more discriminative information needs to be captured. This fact motivated scientists to develop the orthogonal moments, which use as kernel functions polynomials that constitute an orthogonal basis. This property of orthogonality, gives to the corresponding moments the feature of minimum information redundancy, meaning that different moment orders describe different image part. Such moment families are the Legendre [1], Zernike [6], Pseudo-Zernike [1], Fourier–Mellin [14], Tchebichef [15], Krawtchouk [3] moments. These moments can be used as image descriptors after an appropriate normalization procedure in order to achieve translation, scale and rotation invariances.

Among the above orthogonal moment families the Krawtchouk moments exhibit significant properties in comparison with the other ones, due to their high noise robustness and local behaviour and have been selected as the base on which the proposed methodology is applied.

In this paper, a novel methodology that generates moment invariants from a set of Krawtchouk moment invariants (KMIs), which improves the overall classification performance in difficult pattern recognition applications, is presented and analysed in detail.

The proposed methodology is based on a new image representation technique introduced by the authors in [19], which gives more degrees of freedom to the computation of moment invariants. By applying the image slice representation (ISR) technique an image can be decomposed to a finite number of intensity slices, and the computation of the total image moments can be reduced to the computation of the corresponding moment of each image's slices. The resulted moments of each slice, can be considered as the components of the original moment in the intensity domain.

The performance of the newly proposed moment invariants, which are called slice moment invariants (SMIs), are investigated in detail through appropriate experiments to several well known benchmark pattern recognition problems. The experiments show that the conventional KMIs can effectively be replaced by a new set of moment invariants since they perform better in recognizing patterns.

The paper is organized as follows: Section 2 describes the Krawtchouk moment invariants and the way they can be derived. The proposed surrogate moment invariants are introduced in Section 3, while a detailed experimental study demonstrates the efficiency of the novel invariants, in terms of classification performance, in Section 4. Finally, the main conclusions are summarized and discussed in Section 5.

Section snippets

Krawtchouk moment invariants

Krawtchouk orthogonal moments constitute a high discriminative moment family in the discrete domain introduced in image analysis by Yap et al. [3]. These moments are making use of the discrete Krawtchouk polynomials, having the following form, $K_{n} (x; p, N) =_{2} F_{1} (- n, - x; - N; \frac{1}{p}) = \sum_{k = 0}^{N} a_{k, n, p} x^{k}$ where x,n=0,1,2,…,N, N>0, $p \in (0, 1)$ and $_{2} F_{1}$ is the hypergeometric function.

The computation of the Krawtchouk polynomials by using Eq. (1), presents numerical fluctuations and therefore a more stable version of them,

Slice moment invariants

The computation of KMIs by using Eqs. (9), (10), (11), (12), (13), (14) seems to be a time consuming task because of the appearance of quite complicated quantities and repetitive calculations. Additionally, the recursive formulas Eqs. (6), (7), (8) cannot be applied, since the description of KMIs through the GMIs eliminates the occurrence of the original Krawtchouk polynomials that can be quickly computed recursively. The overall overhead in computing the KMIs increases significantly when

Experimental study

In order to investigate the discrimination performance of the proposed moment invariants, four classification problems were arranged. The COIL-20 [21], 53_obj [22] computer vision and the ORL [23] ,Yale [24] face datasets, their main properties are shown in Table 1, were selected as benchmarks for studying the classification capabilities of the SMIs in comparison with the corresponding conventional ones.

Conclusion

A new methodology for the generation of high discriminant moment invariants based on the ISR representation was introduced in the previous sections. The proposed moment invariants are computed as parts of the original moment invariants and they present zero information redundancy since they describe non-overlapping information of the image in process. Despite their high classification capabilities the proposed invariants eliminate the need for high order moment invariant computations, a process

About the Author—GEORGE A. PAPAKOSTAS received the Diploma in Electrical and Computer Engineering in 1999 and the M.Sc. and Ph.D. Degree in Electrical and Computer Engineering (topic in Feature Extraction and Pattern Recognition) in 2002 and 2007, respectively, from Democritus University of Thrace (DUTH), Greece. He is the author of 30 publications in international scientific journals and conferences. His research interests are focused in the field of pattern recognition, neural networks,

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The extraction of chemical information in samples is essential to analytical modeling. In this contribution, the applications of several image moments on the analyses of chemical signals are introduced, including the quantitative analysis of multiple target components in mixture and the qualitative analysis of samples, based on the chemical spectra obtained from HPLC coupled with photodiode array detector, LC-MS, Fluorescence, Terahertz, NMR or NIR analytical instruments. As the extension, chemical image moment approach has also been used in the prediction of protein phosphorylation sites as well as quantitative structure-activity relationship (QSAR) researches. The obtained satisfactory results indicate that the proposed approach is a convenient way to effectively extract target information and could reduce the difficulty of analytical experiments and improve the analytical speed, accuracy and reliability. Our study presents the specific charm of chemical image moments and provides novel insight into analytical chemistry.
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The main purpose of this work is to present a comparison between the classical discrete moment invariants and separable discrete moment invariants based on Racah polynomials in terms 2D object recognition accuracy. For that, we present a pattern recognition system based on separable moment invariants and using the 1-NN (k-Nearest Neighbor) as method of classification. The results show that the separable discrete moment invariants express significant improvement in terms of recognition accuracy and invariability.
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It was first introduced to the image analysis community by Hu [6]. For over the past 50 years, the moment functions have been extensively studied and applied in various areas like image analysis by Žunić et al. [16], pattern recognition [4,8,12,17,28], image watermarking [2,24], image compression [5,21] etc. As problems and new applications are encountered, the moment functions have evolved over time to new variants of moment functions with additional features and functionalities.
Anisotropic scale and translation invariants (ASTI) for Tchebichef moments have been proposed by Zhu et al. [27]. Since these invariants are derived via the decomposition of Tchebichef polynomials, it is unavoidable that the invariant algorithms inherit the complexities from the Tchebichef polynomials defined in terms of hypergeometric functions. Furthermore, in order to achieve anisotropic scale and translation invariance, the computation of translation invariants and scale invariants need to be performed sequentially. These have turned out to be the bottleneck for the invariant algorithms. Experimental results show that some of the computed ASTI features from symmetric patterns are less accurate. Thus, we would like to extend the work of Zhu et al. [27] to simplify the complexity of the algorithms and further improve the accuracy of the computed features. The three terms recurrence relation of the Tchebichef polynomials has been used to simplify and improve the computational efficiency of the invariant algorithms. Skew transformations are deployed to enhance the numerical accuracy of ASTI for Tchebhcief moments. Our studies show that the skewed features are less sensitive to noise and significantly enhance the accuracy of pattern recognition systems. This has been verified by the experiments on recognition of printed English letters and leaf patterns corrupted by noise and scaled and translated deformations. The simplification of the algorithms using the orthogonal property of basis functions also can be used to simplify more complex invariants like affine invariants of discrete Tchebichief moments. It can also be extended to derive invariants for other orthogonal based moments like Legendre moments, Krawtchouk moments, Hahn moments, etc.
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Both continuous orthogonal moments and discrete orthogonal moments can disassemble the information of grayscale image effectively in different levels, and then the information of target components in mixture can be extracted and applied to chemical analysis without using specific methods to remove irrelevant information and interferences. Continuous orthogonal moments (e.g. Wavelet moments) were proposed owing to the advantage of minimal information redundancy and better feature representation ability compared with geometric moments [28–30]. However, the two main drawbacks of the numerical approximation by discrete summation and transformation of the image coordinate space to the domain, which decrease their performance when they are applied to complex systems.
The interference signals and overlapped peaks are common phenomena in fluorescence determination, which seriously influence the accuracy and reliability of analytical results. In this paper, Krawtchouk image moment method was introduced to the analysis of fluorescence three-dimensional (3D) spectra and applied to the quantitative analysis of three drugs including nicotinic acid, metoprolol and amlodipine in human plasma. Without any pretreatment to the obtained spectra, Krawtchouk moments were directly calculated on the grayscale images of 3D spectra, and the quantitative linear models for the three drugs were established by stepwise regression, respectively. The determination coefficients (R) were more than 0.9906. The correlation coefficients of leave-one-out cross-validation (R_cv) were more than 0.9256. The precisions (RSD, %) of inter-day and intra-day variations were less than 8.4% and 3.1%, separately. The recovery was from 101.84% to 107.88%. The limit of quantification and the limit of detection were 0.12 μg mL⁻¹ and 0.43 μg mL⁻¹, respectively. All the statistical parameters supported that the obtained models performed well and the proposed method was accurate and reliable. Our study indicates that Krawtchouk image moments with the powerful abilities of multi-resolution and extracting local information can be applied to the simultaneous determination of multi-target compounds in plasma based on fluorescence 3D spectra.
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To use the discrete orthogonal moments for the object classification, it is indispensable that MTM, MKM and MHM which are invariant under rotation, scaling, and translation of the image. Therefore to obtain the translation, scale and rotation invariants of discrete orthogonal moments, we adopt the same strategy used by Papakostas et al. for Krawtchouk moments [31]. So, we derive the MTMI, MKMI and MHMI invariant moments through the geometric moments using the direct method and the fast method based on the image block representation method.
In this paper, we present a new set of bivariate discrete orthogonal polynomials based on the product of Meixner׳s discrete orthogonal polynomials by Tchebichef׳s, Krawtchouk׳s and Hahn׳s discrete orthogonal polynomials. This set of bivariate discrete orthogonal polynomials is used to define three new types of discrete orthogonal moments as Meixner–Tchebichef moments, Meixner–Krawtchouk moments and Meixner–Hahn moments. We also present an approach to accelerate the computation of these moments by using the image block representation for binary images and image slice representation for gray-scale images. A novel set of Meixner–Tchebichef invariant moments, Meixner–Krawtchouk invariant moments and Meixner–Hahn invariant moments is also derived. These invariant moments are derived algebraically from the geometric invariant moments and their computation is accelerated using an image representation scheme. The proposed algorithms are tested using several well-known computer vision datasets including, moment׳s invariability and pattern recognition. The performance of these invariant moments used as pattern features for a pattern classification is compared with the shape descriptors of Hu, Legendre, Tchebichef–Krawtchouk, Krawtchouk–Hahn and Tchebichef–Hahn invariant moments, the texture descriptors and the color descriptors for four different databases.

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About the Author—KARAKASSIS EVAGGELOS received the Diploma in Production Management Engineering. He is currently pursuing the Ph.D. degree in the Department of Production Management Engineering, Democritius University of Thrace, Greece. His research interest includes robotics, image processing, computational intelligence.

About the Author—DIMITRIOS E. KOULOURIOTIS received his Ph.D. in Intelligent Systems from Department of Production and Management Engineering, Technical University of Crete, Greece, in 2001. He received the Diploma in Electrical and Computer Engineering at the Democritus University of Thrace and the M.S. in Electronic and Computer Engineering at the Technical University of Crete, in 1993 and 1996, respectively. He is currently in the Department of Production and Management Engineering, Democritus University of Thrace, as an Assistant Professor. His research interests include computational intelligence, machine learning and pattern recognition and their applications in vision, image and signal processing, robotics and production engineering, operational research and financial engineering. He has published over 100 research papers in these areas, and he served as reviewer in numerous conferences and journals.

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Novel moment invariants for improved classification performance in computer vision applications

Abstract

Introduction

Section snippets

Krawtchouk moment invariants

Slice moment invariants

Experimental study

Conclusion

Information Sciences

Pattern Recognition Letters

Pattern Recognition

Image and Vision Computing

Applied Mathematics and Computation

Pattern Recognition

Pattern Recognition

Pattern Recognition

Moment Functions in Image Analysis

Image analysis via the general theory of moments

Journal of Optical Society of America

Image analysis by Krawtchouk moments

IEEE Transactions on Image Processing

Visual pattern recognition by moment invariants

IRE Transactions on Information Theory