3D radial invariant of dual Hahn moments

In this work, we propose new sets of 2D and 3D rotation invariants based on orthogonal radial dual Hahn moments, which are orthogonal on a non-uniform lattice. We also present theoretical mathematics to derive them. Thus, this paper presents in the first case new 2D radial dual Hahn moments based on polar representation of an image by one-dimensional orthogonal discrete dual Hahn polynomials and a circular function. The dual Hahn polynomials are general case of Tchebichef and Krawtchouk polynomials. In the second case, we introduce new 3D radial dual Hahn moments employing a spherical representation of volumetric image by one-dimensional orthogonal discrete dual Hahn polynomials and a spherical function, which are orthogonal on a non-uniform lattice. The 2D and 3D rotational invariants are extracts from the proposed 2D and 3D radial dual Hahn moments respectively. In order to test the proposed approach, three problems namely image reconstruction, rotational invariance and pattern recognition are attempted using the proposed moments. The result of experiments shows that the radial dual Hahn moments have performed better than the radial Tchebichef and Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial dual Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image and PSB database for 3D image.

     

Radial Hahn Moment Invariants for 2D and 3D Image Recognition

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional (2D and 3D) image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues: the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark (PSB) database for 3D image.     

Radial invariant of 2D and 3D Racah moments

In this paper, we introduce new sets of 2D and 3D rotation, scaling and translation invariants based on orthogonal radial Racah moments. We also provide theoretical mathematics to derive them. Thus, this work proposes in the first case a new 2D radial Racah moments based on polar representation of an object by one-dimensional orthogonal discrete Racah polynomials on non-uniform lattice, and a circular function. In the second case, we present new 3D radial Racah moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Racah polynomials and a spherical function. Further 2D and 3D invariants are extracted from the proposed 2D and 3D radial Racah moments respectively will appear in the third case. To validate the proposed approach, we have resolved three problems. The 2D/ 3D image reconstruction, the invariance of 2D/3D rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Racah moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges rapidly to the original image using 2D and 3D radial Racah moments, and the test 2D/3D images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and PSB database for 3D image.     

Three dimensional radial Tchebichef moment invariants for volumetric image recognition

The property of rotation, scaling and translation invariant has a great important in 3D image classification and recognition. Tchebichef moments as a classical orthogonal moment have been widely used in image analysis and recognition. Since Tchebichef moments are represented in Cartesian coordinate, the rotation invariance can’t easy to realize. In this paper, we propose a new set of 3D rotation scaling and translation invariance of radial Tchebichef moments. We also present a theoretical mathematics to derive them. Hence, this paper we present a new 3D radial Tchebichef moments using a spherical representation of volumetric image by a one-dimensional orthogonal discrete Tchebichef polynomials and a spherical function. They have better image reconstruction performance, lower information redundancy and higher noise robustness than the existing radial orthogonal moments. At last, a mathematical framework for obtaining the rotation, scaling and translation invariants of these two types of Tchebichef moments is provided. Theoretical and experimental results show the superiority of the proposed methods in terms of image reconstruction capability and invariant recognition accuracy under both noisy and noise-free conditions. The result of experiments prove that the Tchebichef moments have done better than the Krawtchouk moments with and without noise. Simultaneously, the reconstructed 3D image converges quickly to the original image using 3D radial Tchebichef moments and the test images are clearly recognized from a set of images that are available in a PSB database.     

Rotation Scaling and Translation Invariants of 3D Radial Shifted Legendre Moments

This paper proposes a new set of 3D rotation scaling and translation invariants of 3D radially shifted Legendre moments. We aim to develop two kinds of transformed shifted Legendre moments: a 3D substituted radial shifted Legendre moments (3DSRSLMs) and a 3D weighted radial one (3DWRSLMs). Both are centered on two types of polynomials. In the first case, a new 3D radial complex moment is proposed. In the second case, new 3D substituted/weighted radial shifted Legendre moments (3DSRSLMs/3DWRSLMs) are introduced using a spherical representation of volumetric image. 3D invariants as derived from the suggested 3D radial shifted Legendre moments will appear in the third case. To confirm the proposed approach, we have resolved three issues. To confirm the proposed approach, we have resolved three issues: rotation, scaling and translation invariants. The result of experiments shows that the 3DSRSLMs and 3DWRSLMs have done better than the 3D radial complex moments with and without noise. Simultaneously, the reconstruction converges rapidly to the original image using 3D radial 3DSRSLMs and 3DWRSLMs, and the test of 3D images are clearly recognized from a set of images that are available in Princeton shape benchmark (PSB) database for 3D image.     

New set of adapted Gegenbauer–Chebyshev invariant moments for image recognition and classification

Orthogonal moments and their invariants to geometric transformations for gray-scale images are widely used in many pattern recognition and image processing applications. In this paper, we propose a new set of orthogonal polynomials called adapted Gegenbauer–Chebyshev polynomials (AGC). This new set is used as a basic function to define the orthogonal adapted Gegenbauer–Chebyshev moments (AGCMs). The rotation, scaling, and translation invariant property of (AGCMs) is derived and analyzed. We provide a novel series of feature vectors of images based on the adapted Gegenbauer–Chebyshev orthogonal moments invariants (AGCMIs). We practice a novel image classification system using the proposed feature vectors and the fuzzy k-means classifier. A series of experiments is performed to validate this new set of orthogonal moments and compare its performance with the existing orthogonal moments as Legendre invariants moments, the Gegenbauer and Tchebichef invariant moments using three different image databases: the MPEG7-CE Shape database, the Columbia Object Image Library (COIL-20) database and the ORL-faces database. The obtained results ensure the superiority of the proposed AGCMs over all existing moments in representation and recognition of the images.

     

Image representation using separable two-dimensional continuous and discrete orthogonal moments

This paper addresses bivariate orthogonal polynomials, which are a tensor product of two different orthogonal polynomials in one variable. These bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments. Some elementary properties of the proposed continuous Chebyshev–Gegenbauer moments (CGM), Gegenbauer–Legendre moments (GLM), and Chebyshev–Legendre moments (CLM), as well as the discrete Tchebichef–Krawtchouk moments (TKM), Tchebichef–Hahn moments (THM), Krawtchouk–Hahn moments (KHM) are presented. We also detail the application of the corresponding moments describing the noise-free and noisy images. Specifically, the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials. The global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform. Comparisons with the known moments are performed, and the results show that the proposed moments are useful in the field of image analysis. Furthermore, the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation, scale and translation, and an example is given of using the proposed moment invariants as pattern features for a texture classification application.     

Projective invariants of co-moments of 2D images

Functions of moments of 2D images that are invariant under some changes are important in image analysis and pattern recognition. One of the most basic changes to a 2D image is geometric change. Two images of the same plane taken from different viewpoints are related by a projective transformation. Unfortunately, it is well known that geometric moment invariants for projective transformations do not exist in general. Yet if we generalize the standard definition of the geometric moments and utilize some additional information from the images, certain type of projective invariants of 2D images can be derived. This paper first defines co-moment as a moment-like function of image that contains two reference points. Then a set of functions of co-moments that is invariant under general projective transformations is derived. The invariants are simple and in explicit form. Experimental results validated the mathematical derivations.     

Radial shifted Legendre moments for image analysis and invariant image recognition

The rotation, scaling and translation invariant property of image moments has a high significance in image recognition. Legendre moments as a classical orthogonal moment have been widely used in image analysis and recognition. Since Legendre moments are defined in Cartesian coordinate, the rotation invariance is difficult to achieve. In this paper, we first derive two types of transformed Legendre polynomial: substituted and weighted radial shifted Legendre polynomials. Based on these two types of polynomials, two radial orthogonal moments, named substituted radial shifted Legendre moments and weighted radial shifted Legendre moments (SRSLMs and WRSLMs) are proposed. The proposed moments are orthogonal in polar coordinate domain and can be thought as generalized and orthogonalized complex moments. They have better image reconstruction performance, lower information redundancy and higher noise robustness than the existing radial orthogonal moments. At last, a mathematical framework for obtaining the rotation, scaling and translation invariants of these two types of radial shifted Legendre moments is provided. Theoretical and experimental results show the superiority of the proposed methods in terms of image reconstruction capability and invariant recognition accuracy under both noisy and noise-free conditions.     

Translation invariants of Zernike moments

Moment functions defined using a polar coordinate representation of the image space, such as radial moments and Zernike moments, are used in several recognition tasks requiring rotation invariance. However, this coordinate representation does not easily yield translation invariant functions, which are also widely sought after in pattern recognition applications. This paper presents a mathematical framework for the derivation of translation invariants of radial moments defined in polar form. Using a direct application of this framework, translation invariant functions of Zernike moments are derived algebraically from the corresponding central moments. Both derived functions are developed for non-symmetrical as well as symmetrical images. They mitigate the zero-value obtained for odd-order moments of the symmetrical images. Vision applications generally resort to image normalization to achieve translation invariance. The proposed method eliminates this requirement by providing a translation invariance property in a Zernike feature set. The performance of the derived invariant sets is experimentally confirmed using a set of binary Latin and English characters.     

Fast computation of 3D Meixner’s invariant moments using 3D image cuboid representation for 3D image classification

Multimedia Tools and Applications - In this paper, we propose a new fast computation method of 3D discrete orthogonal invariant moments of Meixner. This proposed method is based on two fundamental...     

Image analysis by Bessel-Fourier moments

In this paper, we proposed a new set of moments based on the Bessel function of the first kind, named Bessel-Fourier moments (BFMs), which are more suitable than orthogonal Fourier-Mellin and Zernike moments for image analysis and rotation invariant pattern recognition. Compared with orthogonal Fourier-Mellin and Zernike polynomials of the same degree, the new orthogonal radial polynomials have more zeros, and these zeros are more evenly distributed. The Bessel-Fourier moments can be thought of as generalized orthogonalized complex moments. Theoretical and experimental results show that the Bessel-Fourier moments perform better than the orthogonal Fourier-Mellin and Zernike moments (OFMMs and ZMs) in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions.     

Efficient computation of high-order Meixner moments for large-size signals and images analysis

In this paper, we present an in-depth study on the computational aspects of high-order discrete orthogonal Meixner polynomials (MPs) and Meixner moments (MMs). This study highlights two major problems related to the computation of MPs. The first problem is the overflow and the underflow of MPs values (“Nan” and “infinity”). To overcome this problem, we propose two new recursive Algorithms for MPs computation with respect to the polynomial order n and with respect to the variable x. These Algorithms are independent of all functions that are the origin the numerical overflow and underflow problem. The second problem is the propagation of rounding errors that lead to the loss of the orthogonality property of high-order MPs. To fix this problem, we implement MPs based on the following orthogonalization methods: modified Gram-Schmidt process (MGS), Householder method, and Givens rotation method. The proposed Algorithms for the stable computation of MPs are efficiently applied for the reconstruction and localization of the region of interest (ROI) of large-sized 1D signals and 2D/3D images. We also propose a new fast method for the reconstruction of large-size 1D signal. This method involves the conversion of 1D signal into 2D matrix, then the reconstruction is performed in the 2D domain, and a 2D to 1D conversion is performed to recover the reconstructed 1D signal. The results of the performed simulations and comparisons clearly justify the efficiency of the proposed Algorithms for the stable analysis of large-size signals and 2D/3D images.