Orthogonal Moments Research Articles

Orthogonal Polar V Transforms and application to shape retrieval

The traditional orthogonal moments (e.g., Zernike moments) are formulated with polynomials as their basis that often face the problem of computation difficulty especially with the high-order moments. In this paper, we present a novel set of transforms namely the Polar V Transforms (PVTs). We can use the PVTs not only to generate the rotation-invariant features but also to capture global and local information of images. Since the PVTs basis functions can keep a low order of polynomials, we can significantly speed-up the runtime for computing the kernels. The experimental results have demonstrated that our proposed method outperforms the previous methods in runtimes and achieves very good results in shape retrieval compared to the previous methods especially when the images with high degree of perspective distortions.

Errata and comments on “Errata and comments on Orthogonal moments based on exponent functions: Exponent-Fourier moments”

In this paper, we demonstrate that Improved Exponent–Fourier moments mentioned in Xiao et al. (2015) [1] are with crucial mistakes in its definition, an image cannot be decomposed and reconstructed by these incorrect Improved Exponent–Fourier moments. The corrected definition of Improved Exponent–Fourier moments is thus given in this paper to solve problems about numerical instability involved in the process of calculating the Exponent–Fourier moments and values of radial functions approach infinity at origin. Experimental results show the superiority of the corrected Improved Exponent–Fourier moments.

Image analysis by generalized Chebyshev–Fourier and generalized pseudo-Jacobi–Fourier moments

In this paper, we present two new sets, named the generalized Chebyshev–Fourier radial polynomials and the generalized pseudo Jacobi–Fourier radial polynomials, which are orthogonal over the unit circle. These generalized radial polynomials are then scaled to define two new types of continuous orthogonal moments, which are invariant to rotation. The classical Chebyshev–Fourier and pseudo Jacobi–Fourier moments are the particular cases of the proposed moments with parameter α=0. The relationships among the proposed two generalized radial polynomials and Jacobi polynomials, shift Jacobi polynomials, and the hypergeometric functions are derived in detail, and some interesting properties are discussed. Two recursive methods are developed for computing radial polynomials so that it is possible to improve computation speed and to avoid numerical instability. Simulation results are provided to validate the proposed moment functions and to compare their performance with previous works.

Nonlocal means image denoising using orthogonal moments.

An image denoising method in moment domain has been proposed. The denoising involves the development and evaluation based on the modified nonlocal means (NLM) algorithm. It uses the similarity of the neighborhood, evaluated using Krawtchouk moments. The results of the proposed denoising method have been validated using peak signal-to-noise ratio (PSNR), a well-known quality measure such as structural similarity (SSIM) index and blind/referenceless image spatial quality evaluator (BRISQUE). The denoising algorithm has been evaluated for synthetic and real clinical images contaminated by Gaussian, Poisson, and Rician noise. The algorithm performs well compared to the Zernike based denoising as indicated by the PSNR, SSIM, and BRISQUE scores of the denoised images with an improvement of 3.1 dB, 0.1285, and 4.23, respectively. Further, comparative analysis of the proposed work with the existing techniques has also been performed. It has been observed that the results are competitive in terms of PSNR, SSIM, and BRISQUE scores when evaluated for varying levels of noise.

Image analysis by circularly semi-orthogonal moments

Various types of circularly orthogonal moments have been widely used for image reconstruction and rotation invariant classification. However, they suffer from two errors namely numerical integration error and geometric error, which affect their reconstruction capability and pattern recognition accuracy. In this paper, a novel category of circular moments named circularly semi-orthogonal moments is proposed. In the proposed moment, a set of orthogonal basis functions modulated by a negative power exponential envelope is utilized as the radial basis function. For a given degree n, the radial basis function possesses more compact bandwidth, less cutoff frequency and more zeros compared with the frequently-used circularly orthogonal moments including Zernike and orthogonal Fourier–Mellin moments, and so the circularly semi-orthogonal moment calculated with the zeroth order approximation is more robust to numerical error than the frequently-used circularly orthogonal moments. Furthermore, the capability of the semi-orthogonal moment to describe high spatial frequency components of images is relatively higher than that of the frequently-used circularly orthogonal moments. Experimental results demonstrate that the semi-orthogonal moments calculated with the zeroth order approximation perform better than the frequently-used circularly orthogonal moments in terms of image reconstruction capability and invariant recognition accuracy in noise-free, noisy and smooth distortion conditions. It is also shown that the proposed high order moments are more numerically stable than the circularly orthogonal moments.

Learning Distributions of Image Features by Interactive Fuzzy Lattice Reasoning in Pattern Recognition Applications

This paper describes the recognition of image patterns based on novel representation learning techniques by considering higher-level (meta-)representations of numerical data in a mathematical lattice. In particular, the interest here focuses on lattices of (Type-1) Intervals' Numbers (INs), where an IN represents a distribution of image features including orthogonal moments. A neural classifier, namely fuzzy lattice reasoning (flr) fuzzy-ARTMAP (FAM), or flrFAM for short, is described for learning distributions of INs; hence, Type-2 INs emerge. Four benchmark image pattern recognition applications are demonstrated. The results obtained by the proposed techniques compare well with the results obtained by alternative methods from the literature. Furthermore, due to the isomorphism between the lattice of INs and the lattice of fuzzy numbers, the proposed techniques are straightforward applicable to Type-1 and/or Type-2 fuzzy systems. The far-reaching potential for deep learning in big data applications is also discussed.

MRI texture analysis of GRMD dogs using orthogonal moments: A preliminary study

Magnetic resonance image texture analysis (MRI-TA) has been already demonstrated as a powerful tool to extract image information better than visual examination of grey levels. Human muscular dystrophy is a disease of which diagnosis is rather easy but it remains a lack of atraumatic methods for its follow-up as well as for the management of treatments presently in progress. MRI appears to be a potential tool for disease follow-up and treatment monitoring. The present study is part of the European project MYO-MRI (http://myo-mri.eu/) aimed at improving diagnosis and understanding of muscle pathology. This paper focuses on the use of Legendre and Zernike moment-based texture analysis of MRI T2-weighted images to follow from 2 to 14 months 5 healthy and 5 Golden Retriever Muscular Dystrophy Dogs (GRMD), acknowledged to be the most accurate animal model available for human Duchenne Muscular Dystrophy. Regions-Of-Interest (ROIs) were characterized by moments and the features so extracted were analyzed by Principal Component Analysis and classified by means of Support Vector Machine. The preliminary results show relevant discriminations between dogs populations in reference to disease status and evolution.

A New Object Recognition Approach Using Combination of Texture, Color and Shape Features

This paper reports a new approach for recognizing objects by using combination of texture, color and shape features. Texture features were generated by applying statistical calculation on the image histogram. Color features were computed by using mean, standard deviation, skewness and kurtosis. Shape features were generated using combination of Shen features and basic shapes such as eccentricity and dispersion. The total features were used much less compared to approaches that involve orthogonal moments such as Krawtchouk moments, Zernike moments, or Tchebichef moments. Testing was done by using a dataset that contains 53 kinds of objects. All objects contained in the dataset were various things that can be found in supermarkets or produced by manufacturing. The result shows that the system gave 98.11% of accuracy rate.

Moments and moment invariants in the Radon space

Radon transform has been acknowledged as the promising solution for image processing due to its high noise robustness and the ability of converting the rotation, scaling and translation operations on a pattern image into translations and scaling in the Radon image. Recently, several transforms widely employed in signal processing have been introduced in images׳ Radon space for pattern recognition. However, moments and especially moment invariants in the Radon space have not been thoroughly investigated. In this paper, we introduce a mathematical framework of constructing moments and moment invariants in the Radon space. First, rotational moments which represent non-orthogonal moments and Legendre–Fourier moments which represent orthogonal moments are introduced in the Radon space respectively. On this basis, we propose a method to obtain rotation, scaling and translation as well as affine invariance of these moments in the Radon space. Second, we prove that the proposed moments in the Radon space can be represented by a linear combination of classical geometric moments. With this property, the implementation time of the moments in the Radon space can be significantly reduced, and the recognition accuracy can also be greatly improved since no numerical approximation is involved. Theoretical and experimental analysis on invariant recognition accuracy, noise robustness, image blur distortion and computational time also shows the superiority of the proposed methods.

Prediction of Speech Intelligibility Using a Neurogram Orthogonal Polynomial Measure (NOPM)

Sensorineural hearing loss (SNHL) is an increasingly prevalent condition, resulting from damage to the inner ear and causing a reduction in speech intelligibility. This paper proposes a new speech intelligibility prediction metric, the neurogram orthogonal polynomial measure (NOPM). This metric applies orthogonal moments to the auditory neurogram to predict speech intelligibility for listeners with and without hearing loss. The model simulates the responses of auditory-nerve fibers to speech signals under quiet and noisy conditions. Neurograms were created using a physiologically based computational model of the auditory periphery. A well-known orthogonal polynomial measure, Krawtchouk moments, was applied to extract features from the auditory neurogram. The predicted intelligibility scores were compared to subjective results, and NOPM showed a good fit with the subjective scores for normal listeners and also for listeners with hearing loss. The proposed metric has a realistic and wider dynamic range than corresponding existing metrics, such as mean structural similarity index measure and neurogram similarity index measure, and the predicted scores are also well-separated as a function of hearing loss. The application of this metric could be extended for assessing hearing-aid and speech-enhancement algorithms.

A data-driven smooth test of symmetry

In this paper we propose a data driven smooth test of symmetry. We first transform the raw data via the probability integral transformation according to a symmetrized empirical distribution, and show that under the null hypothesis of symmetry, the transformed data has a limiting uniform distribution, reducing testing for symmetry to testing for uniformity. Employing Neyman’s smooth test of uniformity, we show that only odd-ordered orthogonal moments of the transformed data are required in constructing the test statistic. We present a standardized smooth test that is distribution-free asymptotically and derive the asymptotic behavior of the test and establish its consistency. Extension to dependent data case is discussed. We investigate the finite sample performance of the proposed tests on both homogeneous and mixed distributions (with unobserved heterogeneity). An empirical application on testing symmetry of wage adjustment process, based on heterogeneous wage contracts with different durations, is provided.

Orthogonal moments for determining correspondence between vessel bifurcations for retinal image registration

Retinal image registration is a necessary step in diagnosis and monitoring of Diabetes Retinopathy (DR), which is one of the leading causes of blindness. Long term diabetes affects the retinal blood vessels and capillaries eventually causing blindness. This progressive damage to retina and subsequent blindness can be prevented by periodic retinal screening. The extent of damage caused by DR can be assessed by comparing retinal images captured during periodic retinal screenings. During image acquisition at the time of periodic screenings translation, rotation and scale (TRS) are introduced in the retinal images. Therefore retinal image registration is an essential step in automated system for screening, diagnosis, treatment and evaluation of DR. This paper presents an algorithm for registration of retinal images using orthogonal moment invariants as features for determining the correspondence between the dominant points (vessel bifurcations) in the reference and test retinal images. As orthogonal moments are invariant to TRS; moment invariants features around a vessel bifurcation are unaltered due to TRS and can be used to determine the correspondence between reference and test retinal images. The vessel bifurcation points are located in segmented, thinned (mono pixel vessel width) retinal images and labeled in corresponding grayscale retinal images. The correspondence between vessel bifurcations in reference and test retinal image is established based on moment invariants features. Further the TRS in test retinal image with respect to reference retinal image is estimated using similarity transformation. The test retinal image is aligned with reference retinal image using the estimated registration parameters. The accuracy of registration is evaluated in terms of mean error and standard deviation of the labeled vessel bifurcation points in the aligned images. The experimentation is carried out on DRIVE database, STARE database, VARIA database and database provided by local government hospital in Pune, India. The experimental results exhibit effectiveness of the proposed algorithm for registration of retinal images.

No-Reference Image Blur Assessment Based on Discrete Orthogonal Moments.

Blur is a key determinant in the perception of image quality. Generally, blur causes spread of edges, which leads to shape changes in images. Discrete orthogonal moments have been widely studied as effective shape descriptors. Intuitively, blur can be represented using discrete moments since noticeable blur affects the magnitudes of moments of an image. With this consideration, this paper presents a blind image blur evaluation algorithm based on discrete Tchebichef moments. The gradient of a blurred image is first computed to account for the shape, which is more effective for blur representation. Then the gradient image is divided into equal-size blocks and the Tchebichef moments are calculated to characterize image shape. The energy of a block is computed as the sum of squared non-DC moment values. Finally, the proposed image blur score is defined as the variance-normalized moment energy, which is computed with the guidance of a visual saliency model to adapt to the characteristic of human visual system. The performance of the proposed method is evaluated on four public image quality databases. The experimental results demonstrate that our method can produce blur scores highly consistent with subjective evaluations. It also outperforms the state-of-the-art image blur metrics and several general-purpose no-reference quality metrics.

Fusing magnitude and phase of wavelet moments for content-based image retrieval

The wavelet moments (WMs) are capable to represent the variations in the images as they provide both time and frequency localisation. In this paper the image retrieval system is implemented using complex wavelet moments (CWMs). The magnitude of the WMs of image is rotation invariant, but the phase component changes with image rotation. The magnitude and phase of WMs are incorporated and used both real and imaginary components of WMs so as to make WMs complex to describe as image. The rotation invariance of real and imaginary component is achieved by correction and comparison of phases of WMs of two comparable images with suitable distance metric. The proposed algorithm is applied to USC Rotated Texture database and Colombia Object Image Library (COIL-100) database and detailed analysis is carried out and performance is compared with other orthogonal moments. In many cases the complex WMs produce excellent performance. The average retrieval accuracy for 13 textures, Bark, Brick, Bubble, Grass, Leather, Pigskin, Raffia, Sand, Straw, Water, Weave, Wood and Wool are 100, 85, 92, 99, 91, 92, 95, 97, 76, 94, 99, 74, and 95% respectively. Also, the experimental results on COIL-100 database reveals that the complex wavelet moment outperforms than complex Zernike moments and it is one of the useful descriptor for content-based image retrieval systems.

General form for obtaining unit disc-based generalized orthogonal moments.

The rotation invariance of the classical disc-based moments, such as Zernike moments (ZMs), pseudo-ZMs (PZMs), and orthogonal Fourier-Mellin moments (OFMMs), makes them attractive as descriptors for the purpose of recognition tasks. However, less work has been performed for the generalization of these moment functions. In this paper, four general forms are developed to obtain a class of disc-based generalized radial polynomials that are orthogonal over the unit circle. These radial polynomials are scaled to ensure numerical stability, and some useful properties are discussed for potential applications they could be used in. Then, these scaled radial polynomials are used as kernel functions to construct a series of unit discbased generalized orthogonal moments (DGMs). The variation of parameters in DGMs can form various types of orthogonal moments: 1) generalized ZMs; 2) generalized PZMs; and 3) generalized OFMMs. The classical ZMs, PZMs, and OFMMs correspond to a special case of these three generalized moments for which the free parameter α = 0. Each member of this family will share some excellent properties for image representation and recognition tasks, such as orthogonality and rotation invariance. In addition, we have also developed two algorithms, the so-called m-recursive and n-recursive methods for the computation of these proposed radial polynomials to improve the numerical stability. Experimental results show that the proposed methods are superior to the classical disc-based moments in terms of image representation capability and classification accuracy.

Errata and comments on “Orthogonal moments based on exponent functions: Exponent-Fourier moments”

In this paper, we show that the Exponent-Fourier moments (EFMs), as defined in Ref. [1] are imprecise and the radial function of EFMs performs numerically unstable. At last, an improvement of Exponent-Fourier moments (IEFMs) is given for avoiding the above problems. Experimental results also show the superiority of the proposed methods.

Image analysis using separable discrete moments of Charlier-Hahn

In this paper, we present a new set of bivariate discrete orthogonal polynomials defined from the product of Charlier and Hahn discrete orthogonal polynomials with one variable. This bivriate polynomial is used to define other set of separable two-dimensional discrete orthogonal moments called Charlier-Hahn's moments. We also propose the use of the image slice representation methodology for fast computation of Charlier-Hahn's moments. In this approach the image is decomposed into series of non-overlapped binary slices and each slice is described by a number of homogenous rectangular blocks. Thus, the moments of Charlier-Hahn can be computed fast and easily from the blocks of each slice. A novel set of Charlier-Hahn invariant moments is also presented. These invariant moments are derived algebraically from the geometric invariant moments and their computation is accelerated using an image representation scheme. The presented approaches are tested in several well known computer vision datasets including computational time, image reconstruction, the moment's invariability and the classification of objects. The performance of these invariant moments used as pattern features for a pattern classification is compared with Charlier, Hahn, Tchebichef-Krawtchouk, Tchebichef-Hahn and Krawtchouk-Hahn invariant moments.

Enhancing noisy speech signals using orthogonal moments

This study describes a new approach to enhance noisy speech signals using the discrete Tchebichef transform (DTT) and the discrete Krawtchouk transform (DKT). The DTT and DKT are based on well-known orthogonal moments: the Tchebichef and Krawtchouk moments, respectively. The representations of speech signals using a limited number of moment coefficients and their behaviour in the domain of orthogonal moments are shown. The method involves removing noise from the signal using a minimum-mean-square error in the domain of the DTT or DKT. According to comparisons with traditional methods, the initial experiments yield promising results and show that orthogonal moments are applicable in the field of speech signal enhancement. The application of orthogonal moments could be extended to speech analysis, compression and recognition.

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.

Fast computation of separable two-dimensional discrete invariant moments for image classification

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. New set of bivariate discrete orthogonal polynomials.Separable two-dimensional discrete invariant moments.Improving the performance of image classification.Acceleration of computational time of discrete orthogonal invariant moments.