2D Moments Research Articles

New set of non-separable 2D and 3D invariant moments for image representation and recognition

The extraction of feature vectors of the images is the most important step in image recognition, because it allows a good description of the forms to be recognized. This step is an operation that makes it possible to convert an image into a vector of real or complex values that can serve as a signature for this image. For an accurate recognition system, the used feature vector must be invariant to the three image transformations (translation, rotation and scale), which means that, the descriptor vectors of the image and the transformed image by translation, rotation or scale must be equal. Given the importance of moments and their invariants in pattern recognition and imaging, based on Newton’s binomial and trinomial formulas and the normalized central moments, we construct in this paper four new series of moments for 2D and 3D image recognition, which are invariant to translation, scaling and rotation (TSR): the first is a set of non-orthogonal moment invariants for 2D images (2dMIs), the second is a set of orthogonal moment invariants for 2D images (2dOMIs), the third is a set of non-orthogonal moment invariants for 3D images (3dMIs) and the fourth is a set of orthogonal moment invariants for 3D images (3dOMIs). Using the proposed invariant moments (2dMIs), (2dOMIs), (3dMIs) and (3dOMIs), we construct two types of image descriptor vectors for 2D images and two types for 3D images, which are invariant to translation, rotation and scale. A series of experiments is performed to validate this new set of invariant moments and compare its performance with the existing invariants moments. The obtained results ensure the superiority of the proposed moments over all existing moments in image recognition. Experiments on processing time show that the proposed method is faster than the existing orthogonal invariant moments.

A fast and accurate computation of 2D and 3D generalized Laguerre moments for images analysis

In this paper, we will present a new set of 2D and 3D continuous orthogonal moments based on generalized Laguerre orthogonal polynomials (GLPs) for 2D and 3D image analysis. However, the computation of the generalized Laguerre orthogonal moments (GLMs) is limited by the problems of discretization of the continuous space of the polynomials, approximation of the integrals by finite sums and of too high computation time. To remedy these problems, we will propose a new method for the fast and the precise computation of 2D and 3D GLMs. This method is based on the development of an exact calculation of the double and triple integrals which define the 2D and 3D GLMs, and on the matrix calculation to accelerate the processing time of the images instead of the direct calculation. In addition to the theoretical results obtained, several experiments are carried out to validate the efficiency of the 2D and 3D GLMs descriptors in terms of computation precision and accuracy and in terms of acceleration of computation time and 2D/3D image reconstruction. The experimental results clearly show the advantages and the effectiveness of GLMs compared to the continuous orthogonal moments of Legendre, Chebyshev, Gegenbauer and Gaussian-Hermite.

Tensor trains and moment conservation for multivariate aggregation in population balance modeling

We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. This class of equations is numerically challenging to solve - the computational complexity of “straightforward” algorithms grows exponentially with respect to the number of internal coordinates describing particle properties.Here, we develop algorithms which reduce the storage and computational complexity to almost linear order, O(dn) and O(dnlog⁡(n)), respectively, where d denotes the number of internal coordinates and n the number of pivots per internal coordinate. In particular, we develop fast algorithms in tensor train format to evaluate the multidimensional aggregation integral exploiting fast Fourier transformation for the underlying convolution. A further significant result lies in the conservation of the first 2d moments for our proposed method. Numerical tests confirm the favorable theoretical results concerning computational complexity and conservation of moments.

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

In this paper, we propose a new set of 2D and 3D rotation invariants based on orthogonal radial Meixner moments. We also present a theoretical mathematics to derive them. Hence, this paper introduces in the first case a new 2D radial Meixner moments based on polar representation of an object by a one-dimensional orthogonal discrete Meixner polynomials and a circular function. In the second case, we present a new 3D radial Meixner moments using a spherical representation of volumetric image by a one-dimensional orthogonal discrete Meixner polynomials and a spherical function. Further 2D and 3D rotational invariants are derived from the proposed 2D and 3D radial Meixner moments respectively. In order to prove the proposed approach, three issues are resolved mainly image reconstruction, rotational invariance and pattern recognition. The result of experiments prove that the Meixner moments have done better than the Krawtchouk moments with and without nose. Simultaneously, the reconstructed volumetric image converges quickly to the original image using 2D and 3D radial Meixner moments and the test images are clearly recognized from a set of images that are available in a PSB database.

Simultaneous Recognition and Relative Pose Estimation of 3D Objects Using 4D Orthonormal Moments.

Both three-dimensional (3D) object recognition and pose estimation are open topics in the research community. These tasks are required for a wide range of applications, sometimes separately, sometimes concurrently. Many different algorithms have been presented in the literature to solve these problems separately, and some to solve them jointly. In this paper, an algorithm to solve them simultaneously is introduced. It is based on the definition of a four-dimensional (4D) tensor that gathers and organizes the projections of a 3D object from different points of view. This 4D tensor is then represented by a set of 4D orthonormal moments. Once these moments are arranged in a matrix that can be computed off-line, recognition and pose estimation is reduced to the solution of a linear least squares problem, involving that matrix and the 2D moments of the observed projection of an unknown object. The abilities of this method for 3D object recognition and pose estimation is analytically proved, demonstrating that it does not rely on experimental work to apply a generic technique to these problems. An additional strength of the algorithm is that the required projection is textureless and defined at a very low resolution. This method is computationally simple and shows very good performance in both tasks, allowing its use in applications where real-time constraints have to be fulfilled. Three different kinds of experiments have been conducted in order to perform a thorough validation of the proposed approach: recognition and pose estimation under z axis (yaw) rotations, the same estimation but with the addition of y axis rotations (pitch), and estimation of the pose of objects in real images downloaded from the Internet. In all these cases, results are encouraging, at a similar level to those of state-of-the art algorithms.

Erratum to: Radial invariant of 2D and 3D Racah moments

Erratum to: Radial invariant of 2D and 3D Racah moments

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.

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.

A quantitative description of the morphological aspects of materials structures suitable for quantitative comparisons of 3D microstructures

While several laboratories can produce 3D models of microstructure from serial sectioning or tomography, the more widespread practice is to attempt to infer 3D shapes from 2D sections of different orientations. We have developed a forward modeling approach that, given a 3D particle shape, produces statistically representative 2D sections whose projections are characterized by 2D moment invariants (MI). Since the sectioning plane is random, each particle will produce a statistical distribution of 2D moments and an ensemble of particles in a microstructure will produce its own characteristic distribution of 2D moments and can be used as a representation of the microstructure. This distribution can be represented as a point in a metric space. We use the Hellinger distance as the measure for this space, which allows us to quantify the similarity of two microstructures. Example applications include: determination of a 3D shape by computing the Hellinger distance between MI density maps derived from random 2D section micrographs and the density map database; automated detection and quantification of rafting in cuboidal microstructures; and quantitative comparison of pairs of microstructures.

2D Finite Rate of Innovation Reconstruction Method for Step Edge and Polygon Signals in the Presence of Noise

The finite rate of innovation (FRI) principle is developed for sampling a class of non-bandlimited signals that have a finite number of degrees of freedom per unit of time, i.e., signals with FRI. This sampling scheme is later extended to three classes of sampling kernels with compact support and applied to the step edge reconstruction problem by treating the image row by row. In this paper, we regard step edges as 2D FRI signals and reconstruct them block by block. The step edge parameters are obtained from the 2D moments of a given image block. Experimentally, our technique can reconstruct the edge more precisely and track the Cramér-Rao bounds (CRBs) closely with a signal-to-noise ratio (SNR) larger than 4 dB on synthetic step edge images. Experiments on real images show that our proposed method can reconstruct the step edges under practical conditions, i.e., in the presence of various types of noise and using a real sampling kernel. The results on locating the corners of data matrix barcodes using our method also outperform some state-of-the-art barcode decoders.

The nature of Arctic polar vortices in chemistry–climate models

AbstractThe structure of the Arctic stratospheric polar vortex in three chemistry–climate models (CCMs) taken from the CCMVal‐2 intercomparison is examined using zonal mean and geometric‐based methods. The geometric methods are employed by taking 2D moments of potential vorticity fields that are representative of the polar vortices in each of the models. This allows the vortex area, centroid location and ellipticity to be determined, as well as a measure of vortex filamentation. The first part of the study uses these diagnostics to examine how well the mean state, variability and extreme variability of the polar vortices are represented in CCMs compared to ERA‐40 reanalysis data, and in particular for the UMUKCA‐METO, NIWA‐SOCOL and CCSR/NIES models. The second part of the study assesses how the vortices are predicted to change in terms of the frequency of sudden stratospheric warmings and their general structure over the period 1960–2100.In general, it is found that the vortices are climatologically too far poleward in the CCMs and produce too few large‐scale filamentation events. Only a small increase is observed in the frequency of sudden stratospheric warming events from the mean of the CCMVal‐2 models, but the distribution of extreme variability throughout the winter period is shown to change towards the end of the twentyfirst century. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office

Magnetic ordering in Er3Cu4X4 (X = Si, Ge, Sn)

Magnetic ordering of the orthorhombicEr3Cu4X4 (X = Si, Ge, Sn) system has been studied using both119Sn and 166Er Mössbauer spectroscopy, combined with neutron diffraction. We observe twodistinct ordering events for the erbium moments on the 2d and 4e sites(TN (Er 2d)>TN (Er 4e)) and confirm that the Er 2d moments are larger thanthose on the Er 4e site. Comparison of neutron diffraction with166Er Mössbauer spectroscopy shows that the ordering of the Er 4e moments is far from complete, even at0.7 TN, causing neutron diffraction analysis to severely underestimate the Er 4e moments in theEr3Cu4X4 system.

An Efficient Method for Computation of Legendre Moments

The two-dimensional (2D) and three-dimensional (3D) orthogonal moments are useful tools for 2D and 3D object recognition and image analysis. However, the problem of computation of orthogonal moments has not been well solved because there exist few algorithms that can efficiently reduce the computational complexity. As is well known, the calculation of 2D and 3D orthogonal moments by a straightforward method requires a large number of additions and multiplications. In this paper, an efficient algorithm for computing 2D and 3D Legendre moments is presented. First, a new approach is developed for computing Legendre polynomials with one variable; the corresponding results are then used to calculate 1D Legendre moments. Second, we extend our method to calculating 2D Legendre moments, a more accurate approximation formula when an analog original image is digitized to its discrete form is also discussed, and the relationship between the usual approximation and the new approach is investigated. Finally, an efficient method for computing 3D Legendre moments is developed. As one can see, the proposed algorithm improves the computational efficiency significantly and can be implemented easily for high order of moments.

A fast algorithm to compute the moments of the exponential function for application to semiconductor modelling

Presents a fast numerical method for computing the lowest degree moments of the exponential function for a segment, square or cubic domain. An exponential function with the argument a multivariate polynomial with a maximum degree of one in each single variable is considered. These kinds of integral are encountered in exponential‐fitting finite element methods. For the 1D and 2D cases, a fast method based on the direct evaluation of the moments is used, whereas for the 3D case, a quadrature of 2D moments is considered together with some schemes based on the knowledge of the integrand function in order to improve the computational efficiency.

Modelling of rigid objects by bidimensional moments. Applications to the estimation of 3D rotations

In this paper the moment method has been extended to characterize an object moving in three-dimensional (3D) space. The model evolved is a function of the photometrical characteristics and variations of the projected 2D moments of the object in relation to its various positions. Here, the object is characterized by a set of appropriate moments describing it in relation to image translation and rotation. By using five separate camera images of this object in various positions, the zero-, first- and second-order reference moments could be determined. Experimental results applied to the determination of parameters of a 3D rotation have confirmed the validity of such a method.

Nonlinear opto-electromechanics and photodeformation of optical actuators

Optically driven actuators can introduce remote actuation and control effects without any hard-wire connections. In this study, photostrictive (opto-piezoelectric) characteristics and photodeformation of distributed photostrictive optical actuators are investigated and a parametric study of design parameters is conducted. Photodeformation induced by the photostrictive (opto-piezoelectric) effect (a combination of the photovoltaic effect and the converse piezoelectric effect) is discussed and its two-dimensional (2D) constitutive relations are presented. 2D equivalent control forces and moments induced by the photodeformation effect of distributed actuators are formulated, and system governing equations derived. Static and dynamic applications are discussed, and simulation studies of design parameters are conducted and evaluated.

Statistical pattern recognition using efficient two-dimensional moments with applications to character recognition

A new set of efficient two-dimensional (2D) moments is introduced, which are normalized with respect to standard deviation. They are invariant under rotation, translation and scale of the image, less sensitive to noise and appear to have better classification performance over the existing sets of moments. Application of the proposed technique to optical character recognition is given.

Optical character recognition by the method of moments

An investigation of the use of two-dimensional moments as features for recognition has resulted in the development of a systematic method of character recognition. The method has been applied to six machine-printed fonts. Documents used to test the method contained 24 lines of alphanumeric characters. Before scanning a document to be processed, a training document having the same font must be scanned and stored in memory. Characters on the training document are isolated by contour tracing, and then the 2D moments of each character are computed and stored in a library of feature vectors. The document to be recognized is then scanned, and the 2D moments of its characters are compared with those in the library for classification. In this paper we present the selection of a set of moments that provide good discrimination between characters, the comparison of three classification schemes, the selection of a weighting vector that improves the classification performance, and a series of experiments to determine how the recognition rate is affected by the number of library feature vector sets. Recognition rates between 98.5% and 99.7% have been achieved for all fonts tested.