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Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position and momentum., In the Addendum " Comment on 2-D Covariant Affine Integral Quantization(s)" a few errors have been identified in the article [Adv. Oper. Theory (2020) 5:901-935] and they are corrected. Furthermore, some notations have been modified in order to avoid any confusion

Covariant affine integral quantization of the half-plane $$\mathbb{R} \times \mathbb{R}_ * ^ + $$ is presented.We examine the consequences of different quantizer operators built from weight functions on the half-plane. One of these weights yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both position and momentum vectors. An extension to the phase space for the motion of a particle in the punctured plane and its application to the quantum rotating frame are mentioned.

Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both momentum p and position q., Comment: 36 pages, 10 figures

We develop a novel pose estimation technique for synthetic aperture radar image chips based on the 2-D continuous wavelet transform (CWT). The pose estimator exploits a fast approximation of the 2-D CWT so that its computational complexity is comparable to a principle component analysis (PCA) based approach. By incorporating energy from various scales, the CWT approach is more robust than the PCA method. Experiments over the MSTAR database confirm the robustness of the CWT method.

The behaviour of quantum systems in non-inertial frames is revisited from the point of view of affine coherent state (ACS) quantization. We restrict our approach to the one-particle dynamics confined in a rotating plane about a fixed axis. This plane is considered as punctured due to the existence of the rotation center, which is viewed as a singularity. The corresponding phase space is the affine group of the plane and the ACS quantization enables us to quantize the system by respecting the affine symmetry of the true phase space. Our formulation predicts the appearance of an additional quantum centrifugal term, besides the usual angular momentum one, which prevents the particle to reach the singular rotation center. Moreover it helps us to understand why two different non-inertial Schr\"{o}dinger equations are obtained in previous works. The validity of our equation can be confirmed experimentally by observing the harmonic oscillator bound states and the critical angular velocity for their existence., Comment: 5 pages, no figure, references are added

Wavelet transform theory is an efficient multiscale formalism for analysing local structures. This philosophy, when incorporated within quantum mechanics, demands that there be a naturally corresponding, localized quantization theory, in contrast to the variational formalisms in the literature. Through the recently established equivalency formalism between moment quantization theory and continuous wavelet transform theory (Handy C R and Murenzi R 1998 J. Phys. A: Math. Gen. 31 9897 and Handy C R and Murenzi R 1999 J. Phys. A: Math. Gen. 32 8111), we argue that a new quantization prescription can be defined in which the kinetic energy term is set to zero at the (complex) turning points (or turning hypersurfaces). We establish this, both for one- and two-dimensional systems, and clarify the relevancy of multiscale wavelet analysis in this quantization process.

This paper presents a new group-theoretic perspective for signal and image analysis. It addresses the problem of motion analysis and trajectory determination. The construction exploits the properties of motion parameters to be structured in Lie algebras and Lie groups. The motion models are provided by the group structure which carries an entire theoretical build-up. This theoretical construction is based on a natural association of Lie group representations, minimum-mean-squared-error estimations, and variational principles of optimality. These concepts naturally provide a corresponding association of tools based on continuous wavelet transforms, Kalman filters, and Lagrangians. These tools result in highly parallelizable algorithms based on FFTs, gradients, and dynamic programming. The core of the construction is made of spatiotemporal continuous wavelets that are tuned to the motion parameters to perform motion estimation. The motion parameters consist of scale, orientation, location, velocity, acceler...

The analysis of oriented features in images requires two-dimensional directional wavelets. Among these, we study in detail the class of Cauchy wavelets, which are strictly supported in a (narrow) convex cone in spatial frequency space. They have excellent angular selectivity, as shown by a standard calibration test, and they have minimal uncertainty. In addition, we present a new application of directional wavelets, namely a technique for determining the symmetries of a given pattern with respect to rotations and dilation.

The space of polynomials maps onto itself under affine transformations, . This suggests that a moment reformulation of continuous wavelet transform (CWT) theory (the affine convolution, , of a signal, or wavefunction, ) should lead to significant simplifications in its implementation. We present a comprehensive formalism, with numerical examples, that inextricably links moment quantization (MQ) and CWT theory. For rational fraction potential problems and mother wavelets of the form (Q(x) an appropriate polynomial), MQ permits a more efficient and accurate (in a pointwise convergent sense) CWT implementation; whereas, CWT broadens the scope of applicability for MQ methods, and is its natural extension when a more global approximation is desired. Our formalism also gives one justification for the empirical superiority manifested by previous MQ studies, as compared with dyadic wavelet reconstruction methods. We implement our formalism in the context of the quartic, sextic and octic anharmonic oscillator potentials, and demonstrate the flexibility of the method by treating both the Mexican hat wavelet transform, as well as that based on the mother wavelet .

The space of polynomials is invariant under affine maps. This suggests that a moment based analysis can facilitate a first principles incorporation of continuous wavelet transform (CWT) theory into quantum mechanics. We show that this is indeed the case for a large class of Hamiltonians and mother wavelet functions. We establish the equivalence between moment quantization (MQ) and CWT. By so doing, we clearly demonstrate the inherent multiscale structure of MQ analysis with regards to determining the physical energies and corresponding wavefunctions.

Given a one-dimensional Sturm - Liouville Schrodinger problem with rational polynomial potential, we can generate the continuous wavelet transform (CWT) for its discrete states, thereby permitting the systematic multiscale reconstruction of the corresponding bound-state wavefunction. A key component in this is the use of properly dilated (a) and translated (b) moments, , which readily transform the configuration space Hamiltonian into a finite set of dynamically coupled, linear, first-order differential equations in the dilation-related variable, : The infinite scale problem is readily solved through moment quantization methods and used to generate the moments at all scales. We demonstrate the essentials through the rational fraction potential, , and the Coulomb potential.

The two-dimensional continuous wavelet transform (CWT) is characterized by a rotation parameter, in addition to the usual translations and dilations. This enables it to detect edges and directions in images, provided a directional wavelet is used. First we briefly review the general properties of the 2-D CWT and describe several classes of wavelets, including the directional ones. Then we turn to the problem of wavelet calibration. We show, in particular, how the reproducing kernel may be used for defining and evaluating the scale and angle-resolving power of a wavelet. Finally, we illustrate the usefulness of the scale-angle representation of the CWT on the problem of disentangling a train of damped plane waves.

Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform (CWT) based on the 2D Euclidean group with dilations. In this case, the wavelet transform of a 2D signal (an image) is a function of 4 parameters: two translation parameters b(x), b(y), a rotation angle theta and the usual dilation parameter a. For obvious practical reasons, two of the parameters must be fixed, either (a, theta) or (b(x), b(y)), and the WT visualized as a function of the two other ones. We discuss the general properties of the CWT and apply it, both analytically and graphically, to a number of simple geometrical objects: a line, a square, an angle, etc. For large a, the analysis detects the global shape of the objects, and smaller values of a reveal finer and finer details, in particular edges and contours. If the analyzing wavelet is oriented, like the 2D Morlet wavelet, the transform is extremely sensitive to directions: varying the angle theta uncovers the directional features of the objects, if any. The selectivity of a given wavelet is estimated from its reproducing kernel.

This paper presents a novel motion parameter estimation (ME) algorithm based on the spatio-temporal continuous wavelet transform (CWT). The multidimensional nature of the CWT allows for the definition of a multitude of energy densities by integrating over a subset of the CWT parameter space. Three energy densities are used to estimate motion parameters by sequentially optimizing a state vector composed of velocity, position, and size parameters. This optimization is performed on a frame-by-frame basis allowing the algorithm to track moving objects. The ME algorithm is designed to address real world challenges encountered in the defense industry and traffic monitoring scenarios, such as attaining robust performance in noise and handling obscuration and crossing object trajectories.

We introduce a pose estimation method for SAR imagery using the 2D continuous wavelet transform (CWT). The computational complexity of the new approach is comparable to other image- based approaches such as ones that incorporate principle component analysis (PCA). Using the public domain MSTAR database, we show that the CWT-based method provides a better pose estimate than the PCA method.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The Workshop on Image Processing Applications for the Digital Battlefield was held at Clark Atlanta University, September 14-15, 1998. The workshop focused on the area of image processing applications as applied to the needs of the military. Presentations during the workshop covered topics in image and video compression, image analysis for automatic target detection/recognition, motion analysis, and target tracking. The speakers included distinguished representatives from academic, industry, and military sponsors.

This paper discusses the utility of scale-angle continuous wavelet transform features for object classification. These features are used as input to two algorithms: character recognition and target recognition in FLIR images. The corresponding recognition algorithm is robust against noise and allows data reduction. A comparative study is made between two types of directional wavelets derived from the Mexican hat wavelet and the usual template matching.

In this work, we introduce a detection scheme that is able to identify regions of interest during the intermediate stages of an image formation process for ultra-wideband (UWB) synthetic aperture radar. Traditional detection methods manipulate the data after image formation. However, this approach wastes computational resources by resolving to completion the entire scene including area dominated by benign clutter. As an alternative, we introduce a multiscale focus of attention (FOA) algorithm that processes intermediate radar data from a quadtree-based backprojection image formation algorithm. As the stages of the quadtree algorithm progress, the FOA thresholds a detection statistic that estimates the signal-to-background ratio for increasingly smaller subpatches. Whenever a subpatch fails a detection, the FOA cues the image formation processor to terminate further processing of that subpatch. We demonstrate that the FOA is able to decrease the overall computational load of the image formation process by a factor of two. We also show that the new FOA method provides fewer false alarms than the two-parameter CFAR FOA over a small database of UWB radar data.

The Extended Fractal (EF) feature has been shown to lower the false alarm rate for the focus of attention (FOA) stage of a synthetic aperture radar (SAR) automatic target recognition (ATR) system. The feature is both contrast and size sensitive, and thus, can discriminate between targets and many types of cultural clutter at the earliest stages of the ATR. In this paper we modify the EF feature so that one can 'tune' the size sensitivity to the specific targets of interest. We show how to optimize the EF feature using target chip data from the public MSTAR database. We demonstrate improvements in performance for FOA algorithms that include the new feature by comparing the receiver operating characteristic (ROC) curves for all possible combinations of FOA algorithms incorporating EF, two-parameter CFAR, and variance features. Finally, we perform timing experiments on the fused detector to demonstrate the feasibility for implementation of the detector in a real system.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The project aimed at developing new algorithms for tracking missiles and warheads. The initial theoretical foundation for this effort is multiresolution analysis, in particular, continuous multi-dimensional wavelet theory. In concert with this framework, new classes of motion-model techniques for target tracking were developed.

This paper investigates methods to improve template-based synthetic aperture radar (SAR) automatic target recognition (ATR). The approach utilizes clustering methods motivated from the vector quantization (VQ) literature to search for templates that best represent the signature variability of target chips. The ATR performance using these new templates are compared to the performance using standard templates. For baseline SAR ATR, the templates are generated over uniform angular bins in the pose space. A merge method is able to generate templates that provide a nonuniform sampling of the pose space, and the templates produce modest gains in ATR performance over standard templates.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

This paper irtvesiigaies ihe use of Continuous Wavelei Transform (CWT) feaiires for deeciion of targets inlow resoluiion FLIR imagery. We specifically use ihe CWT features corresponding to the integration of targetf eatures at all relevant scales and orientations. These features are combined with non-linear transformations

In this work, we evaluate the robustness of template matching schemes for automatic target recognition (ATR) against the effects of clutter layover. The results of our experiments indicate the performance of template matching ATR in various image transform domains against the signal to clutter ratio (SCR). The purpose of these transforms is to enhance the target features in a chip while suppressing features representative of background clutter or simple noise. The ATR experiments were performed for synthetic aperture radar imagery using target chips in the public domain MSTAR database. The transforms include pointwise nonlinearities such as the logarithm and power operations. The templates are generated using the training portion of the MSTAR database at the nominal SCR. Many different ATR parameterizations are considered for each transform domain where templates are built to represent different ranges of aspect angles in uniform angular bins of 5, 10, 15, 30, and 45 degree increments. The different ATRs were evaluated using the testing portion of the database where synthetic clutter was added to lower the SCR.

Two texture-based and one amplitude-based features are evaluated as detection statistics for synthetic aperture radar (SAR) imagery. The statistics include a local variance, an extended fractal, and a two-parameter CFAR feature. The paper compares the effectiveness of focus of attention (FOA) algorithms that consist of any number of combinations of the three statistics. The public MSTAR database is used to derive receiver-operator-characteristic (ROC) curves for the different detectors at various signal-to-clutter rations (SCR). The database contains one foot resolution X-band SAR imagery. The results in the paper indicate that the extended fractal statistic provides the best target/clutter discrimination, and the variance statistic is the most robust against SCR. In fact, the extended fractal statistic combines the intensity difference information used also by the CFAR feature with the spatial extent of the higher intensity pixels to generate an attractive detection statistics.

In images, anomalies such as edges or object boundaries take on a perceptual significance that is far greater than their numerical energy contribution to the image. Wavelet transform highlights these anomalies by representing them with significant coefficients. The contribution of a wavelet coefficient to the perceptual quality of the image is related to its magnitude. Degradation in image quality due to image compression reflects in the form of reduction in the magnitude of the wavelet coefficients. Since, significant wavelet coefficients appear across different scales and orientations, it is important to observe the wavelet transform at different scales and orientations. In this paper, the wavelet transform of a given image and the reconstructed images at various quality levels are represented in the form of energy density plots suggested in reference one. A quality metric is proposed based on the absolute difference between the energy densities corresponding to the original and reconstructed images. Preliminary results obtained using the scale-based image quality evaluation strategy are reported.

In this paper we address the problem of target tacking when flare decoys and severe noise are present in the acquired sensor data. The input imagery is assumed to be obtained from an optical sensor mounted on the interceptor missile, where the goal of the interceptor is to neutralize the target. For this purpose algorithms running on on-board processors must extract information from the input imagery in order to steer the interceptor's course toward the target. Two challenging cases are considered here. First, the input imagery is assumed to be corrupted by additive Gaussian nose. Here we see, to determine the usefulness of our approach when low cost poor quality sensors are employed for acquisition. Second, scenarios where standard flare decoys are released by the target aircraft are considered, which represent a challenge due to the disparity in intensity of the target aircraft versus the decoys. Results using synthetically generated test sequences are presented.

The two-dimensional continuous wavelet transform (CWT) is characterized by a rotation parameter, in addition to the usual translations and dilations. The CWT has been interpreted as space-frequency representation of two-dimensional signals, where the translation corresponds to the position variable, and the inverse of the scale and the rotation, taken together, correspond to the spatial-frequency variable. The integral of the CWT's squared modulus, with respect to all variables, gives the energy of the original signal. Therefore, an integration on a subset of the parameters gives an energy density in the remaining variables. This paper deals with the implementation of the two basic densities, that is, the position (or aspect-angle) and scale-angle densities.

The increase in the number of multimedia databases consisting of images has created a need for a quick method to search these databases for a particular type of image. An image retrieval system will output images from the database similar to the query image in terms of shape, color, and texture. For the scope of our work, we study the performance of multiscale Hurst parameters as texture features for database image retrieval over a database consisting of homogeneous textures. These extended Hurst features represent a generalization of the Hurst parameter for fractional Brownian motion (fBm) where the extended parameters quantize the texture roughness of an image at various scales. We compare the retrieval performance of the extended parameters against traditional Hurst features and features obtained from the Gabor wavelet. Gabor wavelets have previously been suggested for image retrieval applications because they can be tuned to obtain texture information for a number of different scales and orientations. In our experiments, we form a database combining textures from the Bonn, Brodatz, and MIT VisTex databases. Over the hybrid database, the extended fractal features were able to retrieve a higher percentage of similar textures than the Gabor features. Furthermore, the fractal features are faster to compute than the Gabor features.