Legendre Moments Research Articles

Image Analysis with Legendre Moment Descriptors

In this research, a numerical integration method is proposed to improve the computational accuracy of Legendre moments. To clarify the improved computation scheme, image reconstructions from higher order of Legendre moments, up to 240, are conducted. With the more accurately generated moments, the distributions of image information in a finite set of Legendre moments are investigated. We have concluded that each individual finite set of Legendre moments will represent the unique image features independently, while the even orders of Legendre moments describe most of the image characteristics.

New analytic function expansion nodal (AFEN) method for solving multigroup neutron simplified P3 (SP3) equations

In this study, a new analytic function expansion nodal (AFEN) method was developed to solve multi-group and three dimensional neutron simplified P3 equations (SP3) in reactor cores with rectangular fuel assemblies. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group and moment. The nodes are coupled through the surface averaged partial currents at each nodal interface. Thus, six boundary conditions at each group and Legendre moments have been considered. Coarse group rebalancing (CGR) method was applied to increase the speed of code calculations. The code takes few-groups cross sections produced by a lattice code such as WIMS and calculates the effective multiplication factor, zeroth and second moments of the flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. The numerical results for different benchmark problems demonstrate that solution of SP3 equations by our AFEN method improves both effective multiplication factor (keff) and power distribution compared to our AFEN diffusion method, especially in heterogeneous geometry and mixed-oxide (MOX) fuel problems.

Blocking artifact removal using partial overlapping based on exact Legendre moments computation

In this paper, we present the design of a partial overlapping block using exact Legendre moment computation (POBRELM) for gray-level image reconstruction. We address and solve the problem of artifact caused by independent reconstruction of the block affecting the visual image quality. This new approach takes advantage of only partial information of the neighbors of each block in spite of using global overlapping. Processing less neighborhood area during the exact computation of Legendre moments leads to significant improvement in the processing time. Simulation results show that the proposed method achieves not only considerable enhancement in terms of reconstruction error, but also drastic reduction in computation time. This makes our method attractive for real-time applications.

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.

Legendre moments as high performance bone biomarkers: computational methods and GPU acceleration

We investigate the use of Legendre moments as biomarkers for an efficient and accurate classification of bone tissue on images coming from stem cell regeneration studies. Legendre moments are analysed from three different perspectives: (1) their discriminant properties in a wide set of preselected vectors of features based on our clinical and computational experience, providing solutions whose accuracy exceeds 90%; (2) the amount of information to be retained when using principal component analysis to reduce the dimensionality of the problem to either 2, 3, 4, 5 or 6 dimensions and (3) the use of the -k-feature set problem to identify a k = 4 number of features which are more relevant to our analysis from a combinatorial optimisation approach. These techniques are compared in terms of computational complexity and classification accuracy to assess the strengths and limitations of the use of Legendre moments. The second contribution of this work goes to reduce the computational complexity by using graphics processing units (GPUs) and compute unified device architecture programming [Nvidia Developer Zone. 2014. CUDA. Available from: http:developer.nvidia.comobjectcuda.html]. We exploit single instruction multiple data parallelism and memory bandwidth on GPUs to accelerate the process to more than a 5 × factor on each 64 × 64 image tile. The result is a powerful combination of software and hardware methods which deliver a much faster response in contrast with existing solutions coming from conventional programming on multicore CPU platforms, thus offering a high performance alternative in clinical practice for real-time imaging.

Image classification using a new set of separable two-dimensional discrete orthogonal invariant moments

We propose a new set of bivariate discrete orthogonal polynomials, which are the product of Charlier’s discrete orthogonal polynomials with one variable by Tchebichef, Krawtchouk, and Hahn discrete orthogonal polynomials with one variable. This set of bivariate discrete orthogonal polynomials is used to define several new types of discrete orthogonal moments such as Charlier-Tchebichef moments (CTM), Charlier-Krawtchouk moments (CKM), and Charlier-Hahn moments (CHM). We also present an approach for fast computation of CTM, CKM, and CHM discrete orthogonal moments using the image block representation for binary images and image slice representation for grayscale images. A set of Charlier-Tchebichef invariant moments, Charlier-Krawtchouk invariant moments, and 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 algorithms are tested in several well-known computer vision datasets including image reconstruction, computational time, moment’s invariability, and classification of objects. The performance of these invariant moments used as pattern features for a pattern classification is compared with Hu, Legendre, Tchebichef-Krawtchouk, Tchebichef-Hahn, and Krawtchouk-Hahn invariant moments.

Recognition of Isolated Printed Tifinagh Characters

of the reported works in the field of character recognition systems achieve modest results by using a single method for calculating the parameters of the character image and a single approach in the classification phase of the system. So, in order to improve the recognition rate, this document proposes an automatic system to recognize isolated printed Tifinagh characters by using a fusion of some classifiers and a combination of some features extraction methods. The Legendre moments, Zernike moments, Hu moments, Walsh transform, GIST and texture are used as descriptors in the features extraction phase due to their invariance to translation, rotation and scaling changes. In the classification phase, the neural network, the Bayesian network, the multiclass SVM (Support Vector Machine) and the nearest neighbour classifiers are combined together. The experimental results of each single features extraction method with each single classification method are compared with our approach to show its robustness. A recognition rate of 100 % is achieved by using some combined descriptors and classifiers.

MEASUREMENT OF DIFFERENTIAL CROSS SECTIONS OF p(e, e′π+)n FOR HIGH-LYING RESONANCES AT Q2 < 5 GeV2

The exclusive electroproduction process ep → e′nπ+ was measured in the range of the invariant mass for nπ+ system 1.6 GeV ≤ W ≤ 2.0 GeV, and the photon virtuality 1.8 GeV2 ≤ Q2 ≤ 4.0 GeV2 using CLAS. For the first time, these kinematics are probed in exclusive π+ production from the protons with nearly full coverage in the azimuthal and polar angles of the nπ+ center-of-mass system. In this experiment, approximately 39,000 differential cross-section data points were measured. In this conference proceeding, preliminary results of our latest analysis work are presented on differential cross sections and structure functions as well as Legendre Moments.

Imaging photoelectron circular dichroism of chiral molecules by femtosecond multiphoton coincidence detection

Here, we provide a detailed account of novel experiments employing electron-ion coincidence imaging to discriminate chiral molecules. The full three-dimensional angular scattering distribution of electrons is measured after photoexcitation with either left or right circular polarized light. The experiment is performed using a simplified photoelectron-photoion coincidence imaging setup employing only a single particle imaging detector. Results are reported applying this technique to enantiomers of the chiral molecule camphor after three-photon ionization by circularly polarized femtosecond laser pulses at 400 nm and 380 nm. The electron-ion coincidence imaging provides the photoelectron spectrum of mass-selected ions that are observed in the time-of-flight mass spectra. The coincident photoelectron spectra of the parent camphor ion and the various fragment ions are the same, so it can be concluded that fragmentation of camphor happens after ionization. We discuss the forward-backward asymmetry in the photoelectron angular distribution which is expressed in Legendre polynomials with moments up to order six. Furthermore, we present a method, similar to one-photon electron circular dichroism, to quantify the strength of the chiral electron asymmetry in a single parameter. The circular dichroism in the photoelectron angular distribution of camphor is measured to be 8% at 400 nm. The electron circular dichroism using femtosecond multiphoton excitation is of opposite sign and about 60% larger than the electron dichroism observed before in near-threshold one-photon ionization with synchrotron excitation. We interpret our multiphoton ionization as being resonant at the two-photon level with the 3s and 3p Rydberg states of camphor. Theoretical calculations are presented that model the photoelectron angular distribution from a prealigned camphor molecule using density functional theory and continuum multiple scattering X alpha photoelectron scattering calculations. Qualitative agreement is observed between the experimental results and the theoretical calculations of the Legendre moments representing the angular distribution for the two enantiomers. The electron-ion coincidence technique using multiphoton ionization opens new directions in table-top analytical mass-spectrometric applications of mixtures of chiral molecules.

A Unified Methodology for Computing Accurate Quaternion Color Moments and Moment Invariants

In this paper, a general framework for computing accurate quaternion color moments and their corresponding invariants is proposed. The proposed unified scheme arose by studying the characteristics of different orthogonal polynomials. These polynomials are used as kernels in order to form moments, the invariants of which can easily be derived. The resulted scheme permits the usage of any polynomial-like kernel in a unified and consistent way. The resulted moments and moment invariants demonstrate robustness to noisy conditions and high discriminative power. Additionally, in the case of continuous moments, accurate computations take place to avoid approximation errors. Based on this general methodology, the quaternion Tchebichef, Krawtchouk, Dual Hahn, Legendre, orthogonal Fourier-Mellin, pseudo Zernike and Zernike color moments, and their corresponding invariants are introduced. A selected paradigm presents the reconstruction capability of each moment family, whereas proper classification scenarios evaluate the performance of color moment invariants.

A New Class of Orthogonal Spline Moments and Its Application

This paper introduces a new set of orthogonal moment Functions-franklin Moments (FMs). The kernel functions of FMs is Franklin set, which is a class of complete orthogonal spline function set of degree 1. The implementation of FMs does not involve any numerical approximation and has a rather low computation complexity, since the basis set has the advantages of lower order. These properties make FMs superior to the conventional orthogonal moments such as Legendre moments and Zernike moments, in terms of the image reconstruction. Our simulation results also show that FMs have a better feature representation capability.

Sliced Fluorescence Imaging Techniques for the Study of Photo‐initiated Dynamic Processes in Bulbs

AbstractTo study photo‐initiated dynamic processes in bulbs, sliced fluorescence imaging techniques have been developed to obtain quantum‐state resolved information on the speed and angular distributions of reaction products, and their angular momentum polarization characteristics. For fluorescent chemical species, an unconventional experimental arrangement can generate a two‐dimensional projection of the Newton sphere from a single photolysis center. Consequently, image blurring for photo‐initiated experiments in bulbs can be dramatically diminished. A combination of optical‐optical (or infrared‐optical) double resonance excitation scheme and the fluorescence imaging detection method warrants a novel technique to acquire three‐dimensional sliced fluorescence images of scattering products. A theoretical framework, which employs a double Legendre moment analysis on the central sliced images, has been developed to determine differential cross sections and kinetic energy release of co‐products in photo‐initiated dynamic processes. Experimental images of collisional relaxations of CN photofragments from disequilibrium to equilibrium are presented to illustrate the distinctive advantage of the present method in comparison with other imaging and spectroscopic techniques.

Efficient data partitioning for the GPU computation of moment functions

In our previous work, we have provided tools for an efficient characterization of biomedical images using Legendre and Zernike moments, showing their relevance as biomarkers for classifying image tiles coming from bone tissue regeneration studies (Ujaldón, 2009) [24]. As part of our research quest for efficiency, we developed methods for accelerating those computations on GPUs (Martín-Requena and Ujaldón, 2011) [10,9]. This new stage of our work focuses on the efficient data partitioning to optimize the execution on many-cores and clusters of GPUs to attain gains up to three orders of magnitude when compared to the execution on multi-core CPUs of similar age and cost using 1 Mpixel images. We deploy a successive and successful chain of optimizations which exploit symmetries in trigonometric functions and access patterns to image pixels which are effectively combined with massive data parallelism on GPUs to enable (1) real-time processing for our set of input biomedical images, and (2) the use of high-resolution images in clinical practice.

Colored bosons on top FBA and angular cross section for [formula omitted] production

With full data set that corresponds to an integrated luminosity of 9.4 fb−1, CDF has updated the top-quark forward–backward asymmetry (FBA) as functions of rapidity difference |Δy| and tt¯ invariant mass Mtt¯. Beside the sustained inconsistency between experiments and standard model (SM) predictions at large |Δy| and Mtt¯, an unexpected large first Legendre moment with a1=0.39±0.108 is found. In order to solve the large top FBA, we study the contributions of color triplet scalar and color octet vector boson. We find that the top FBA at |Δy|>1 (Mtt¯>450 GeV) in triplet and octet model could be enhanced to be around 40% (30%) and 26% (20%), whereas the first Legendre moment is a1Di=0.38 and a1Axi=0.23, respectively.

Automatic recognition of radar signals based on time-frequency image shape character

Radar signal recognition is one of the key technologies of modern electronic surveillance systems. Timefrequency image provides a new way for recognizing the radar signal. In this paper, a series of image processing methods containing image enhancement, image threshold binarization and mathematical morphology is utilized to extract the shape character of smoothed pseudo wigner-ville time-frequency distribution of radar signal. And then the identification of radar signal is realized by the character. Simulation results of eight kinds of typical radar signal demonstrate that when signal noise ratio (SNR) is greater than -3 dB, the Legendre moments shape character of the time-frequency image is very stable. Moreover, the recognition rate by the character is more than 90 per cent except for the FRANK code signal when SNR > -3 dB. Test also show that the proposed method can effectively recognize radar signal with less character dimension through compared with exitsing algorithms.

GEOMETRIC ATTACK RESISTANT ROBUST IMAGE WATERMARKING SCHEME

A new watermarking approach based on affine Legendre moment invariants (ALMIs) and local characteristic regions (LCRs) which allows watermark detection and extraction under affine transformation attacks is presented in this paper. It is a non-blind watermarking scheme. Original image color image is converted into HSV color space and divided into four parts. LCR is constructed and a set of affine invariants are derived on LCRs based on Legendre moments for each part. These invariants can be used for estimating the affine transform coefficients on the LCRs. ALMIs are used for watermark embedding, detection and extraction as they provide synchronization and invariant feature which is necessary for a robust watermarking scheme. The proposed scheme shows resistance to geometric distortion, cropping, filtering, compression, and additive noise than the existing ALMI based scheme [Alghoniemy, M. and Tewfik, A. H. [2004] "Geometric invariance in image watermarking," IEEE Trans. Image Process13(2), 145–153] and affine geometric moment invariant (AGMI) based scheme [Seo, J. S. and Yoo, C. D. [2006] "Image watermarking based on invariant regions of scale-space representation," IEEE Trans. Signal Process. 54(4), 1537–1549].

Statistical Model of Shape Moments with Active Contour Evolution for Shape Detection and Segmentation

Abstract This paper describes a novel method for shape representation and robust image segmentation. The proposed method combines two well known methodologies, namely, statistical shape models and active contours implemented in level set framework. The shape detection is achieved by maximizing a posterior function that consists of a prior shape probability model and image likelihood function conditioned on shapes. The statistical shape model is built as a result of a learning process based on nonparametric probability estimation in a PCA reduced feature space formed by the Legendre moments of training silhouette images. A greedy strategy is applied to optimize the proposed cost function by iteratively evolving an implicit active contour in the image space and subsequent constrained optimization of the evolved shape in the reduced shape feature space. Experimental results presented in the paper demonstrate that the proposed method, contrary to many other active contour segmentation methods, is highly resilient to severe random and structural noise that could be present in the data.

Color, texture and shape descriptor fusion with Bayesian network classifier for automatic image annotation

Due to the large amounts of multimedia data prevalent on the Web, Some images presents textural motifs while others may be recognized with colors or shapes of their content. The use of descriptors based on one’s features extraction method, such as color or texture or shape, for automatic image annotation are not efficient in some situations or in absence of the chosen type. The proposed approach is to use a fusion of some efficient color, texture and shape descriptors with Bayesian networks classifier to allow automatic annotation of different image types. This document provides an automatic image annotation that merges some descriptors in a parallel manner to have a vector that represents the various types of image characteristics. This allows increasing the rate and accuracy of the annotation system. The Texture, color histograms, and Legendre moments, are used and merged respectively together in parallel as color, texture and shape features extraction methods, with Bayesian network classifier, to annotate the image content with the appropriate keywords. The accuracy of the proposed approach is supported by the good experimental results obtained from ETH-80 databases.

Invariant Descriptors and Classifiers Combination for Recognition of Isolated Printed Tifinagh Characters

In order to improve the recognition rate, this document proposes an automatic system to recognize isolated printed Tifinagh characters by using a fusion of 3 classifiers and a combination of some features extraction methods. The Legendre moments, Zernike moments and Hu moments are used as descriptors in the features extraction phase due to their invariance to translation, rotation and scaling changes. In the classification phase, the neural network, the multiclass SVM (Support Vector Machine) and the nearest neighbour classifiers are combined together. The experimental results of each single features extraction method and each single classification method are compared with our approach to show its robustness.