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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.

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.

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.

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 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 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.

This paper presents new results on the tracking of ballistic missiles warheads using spatio-temporal wavelets. Here we focus our attention on handling more general classes of motion, such as acceleration. To accomplish this task the spatio-temporal wavelet transform is adapted to the motion parameters on a frame-by-frame basis. Three different energy densities, associated with velocity, location, and size, have been defined to determine motion parameters. We pointed out that maximizing these energy densities is equivalent to a minimum squared error estimation. Tracking results on synthetically generated image sequences demonstrate the capabilities of the proposed algorithm.

The 2-D continuous wavelet transform (CWT) has been used by a number of authors, in a wide variety of physical problems.1 - 3 In all cases, its main purpose is the analysis of images, that is, the detection of specific features such as hierarchical structures or particular discontinuities, edges, filaments, contours, boundaries between areas of different luminosity, etc. Of course, the type of wavelet chosen depends on the precise aim. In fact, the 2-D CWT is based on the 2-D dimensional Euclidean group with dilations, the scr-called twcr-dimensional similitude group, SIM(2).4 It is the purpose of this paper to explore this connection further and draw some of its practical consequences.

(determining E and the initial moment values p,,(i)),and procede to numerically integrate the coupled first order equations. For the class of wavelet functions beingconsidered, the wavelet transform,W'I', is a (finite) linear superposition of the moments and therefore triviallyobtainable. Reconstruction of the corresponding (wavefunction) solution, 'I', ensues by either using well knowndyadic wavelet reconstruction methods, or evaluating the a —

This paper addresses the problem of tracking a ballistic missile warhead. In this scenario, the ballistic missile is assumed to be fragmented into many pieces. The goal of the algorithm presented here is to track the warhead that is among the fragments. It is assumed that images are acquired from an optical sensor located in the interceptor nose cone. This imagery is used by the algorithm to steer the course of interception. The algorithm proposed in this paper is based on continuous spatio-temporal wavelet transforms (CWTs). Two different energy densities of the CWT are used to perform velocity detection and filtering. Additional post-processing is applied to discriminate among objects traveling at similar velocities. Particular attention is given to achieving robust performance on noisy sensor data and under conditions of temporary occlusions. First we introduce the spatio-temporal CWT and stress the relationships with classical orientation filters. Then we describe the CWT- based algorithm for target tracking, and present results on synthetically generated sequences.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The work described in this paper addresses the use of the four-dimensional continuous wavelet transform (CWT) for automatic target recognition (ATR) and detection. This transform is an overcomplete representation with four coordinates: two spatial, t1 and t2; a rotational coordinate, (theta) ; and a scale coordinate, a. Two central ideas are discussed in connection with the transform's application to target recognition. The first is cross-scale reconstruction, which refers to exploiting the dominate presence of target features across scales. The second is utilizing the non-spatial coordinate space as a working environment for feature extraction and classification. This aspect is unique to the multidimensional wavelet transform, emanating from the inherent redundancy in the transform representation. Some conclusions are drawn in the last section regarding the utility of the CWT for ATR, and the transform's potential as an analysis tool.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Wavelet analysis is a particular time- or space-scale representation of signals which has found a wide range of applications in physics and mathematics in the last few years. In order to understand its success, let us consider first the case of one-dimensional signals. As a matter of fact, most real life signals are nonstationary. They often contain transient components, sometimes physically significant, and mostly cover a wide range of frequencies. In addition, there is frequently a direct correlation between the characteristic frequency of a given segment of the signal and the time duration of that segment. Low frequency pieces tend to last a long interval, whereas high frequencies occur in general for a short moment only. Human speech signals are typical in this respect. Vowels have a relatively low mean frequency and last quite long, whereas consonants contain a wide spectrum, especially in the attack, and are often very short.

Automatic target detection and recognition (ATR) requires the ability to optimally extract the essential features of an object from (usually) cluttered environments. In this regard, efficient data representation domains are required in which the important target features are both compactly and clearly represented, enhancing ATR. Since both detection and identification are important, multidimensional data representations and analysis techniques, such as the continuous wavelet transform (CWT), are highly desirable. First we review some relevant properties of two 2D CWT. Then we propose a two-step algorithm based on the 2D CWT and discuss its adequacy for solving the ATR problem. Finally we apply the algorithm to various images.

Both in 1D (signal analysis) and 2D (image processing), the wavelet transform (WT) has become by now a standard tool. Although the discrete version, based on multiresolution analysis, is probably better known, the continous WT (CWT) plays a crucial role for the detection and analysis of particular features in a signal, and we will focus here on the latter. In 2D however, one faces a practical problem. Indeed, the full parameter space of the wavelet transform of an image is 4D. It yields a representation of the image in position parameters (range and perception angle), as well as scale and anisotropy angle. The real challenge is to compute and visualize the full continuous wavelet transform in all four variables--obviously a demanding task. Thus, in order to obtain a manageable tool, some of the variables must be frozen. In other words, one must limit oneself to sections of the parameter space, usually 2D or 3D. For 2D sections, two variables are fixed and the transform is viewed as a function of the two remaing ones, and similarly for 3D sections. Among the six possible 2D sections, two play a privileged role. They yield respectively the position representation, which is the standard one, and the scale-angle representation, which has been proposed and studied systematically by two of us in a number of works. In this paper we will review these results and investigate the four remaining 2D representations. We will also make some comments on possible applications of 3D sections. The most spectacular property of the CWT is its ability at detecting discontinuities in a signal. In an image, this means in particular the sharp boundary between two regions of different luminosity, that is, a contour or an edge. Even more prominent in the transform are the corners of a given contour, for instance the contour of a letter. In a second part, we will exploit this property of the CWT and describe how one may design an algorithm for automatic character recognition (here we obviously work in the position--range-perception angle--representation). Several examples will be exhibited, illustrating in particluar the robustness of the method in the presence of noise.

When one wants to extend to more than one dimension, the whole wavelet machinery developped for the one dimensional ax+b group, while keeping the group language, it is natural to consider the n-dimensional Euclidean group with dilations, to be denoted by IG(n). It is a non-unimodular locally compact group and its most natural unitary representation of in L(ℝn, dn x), is irreducible and square integrable.

Accurate object tracking is important in defense applications where an interceptor missile must hone into a target and track it through the pursuit until the strike occurs. The expense associated with an interceptor missile can be reduced through a distributed processing arrangement where the computing platform on which the tracking algorithm is run resides on the ground, and the interceptor need only carry the sensor and communications equipment as part of its electronics complement. In this arrangement, the sensor images are compressed, transmitted to the ground, and decompressed to facilitate real-time downloading of the data over available bandlimited channels. The tracking algorithm is run on a ground-based computer while tracking results are transmitted back to the interceptor as soon as they become available. Compression and transmission in this scenario introduce distortion. If severe, these distortions can lead to erroneous tracking results. As a consequence, tracking algorithms employed for this purpose must be robust to compression distortions. In this paper we introduced a robust object tracking algorithm based on the continuous wavelet transform. The algorithm processes image sequence data on a frame-by-frame basis, implicitly taking advantage of temporal history and spatial frame filtering to reduce the impact of compression artifacts. Test results show that tracking performance can be maintained at low transmission bit rates and can be used reliably in conjunction with many well-known image compression algorithms.

The two-dimensional (2-D) continuous wavelet transform (CWT) is characterized by a rotation parameter, in addition to the usual translations and dilations. This enables is to detect edges and directions in images, provided a directional wavelet is used. First we review the general properties of the 2-D CWT and describe several useful representations. We describe various classes of wavelets, including the directional ones. Then we turn to the problem of wavelet calibration, in particular, the evaluation of the scale and angle resolving power of a wavelet. Finally we discuss several applications of directional wavelets. (C) 1996 John Wiley & Sons, Inc.

This paper addresses the problem of detecting and tracking moving objects in digital image sequences. The main goal is to detect and select mobile objects in a scene, construct the trajectories, and eventually reconstruct the target objects or their signatures. It is assumed that the image sequences are acquired from imaging sensors. The method is based on spatio-temporal continuous wavelet transforms, discretized for digital signal analysis. It turns out that the wavelet transform can be used efficiently in a Kalman filtering framework to perform detection and tracking. Several families of wavelets are considered for motion analysis according to the specific spatio-temporal transformation. Their construction is based on mechanical parameters describing uniform motion, translation, rotation, acceleration, and deformation. The main idea is that each kind of motion generates a specific signal transformation, which is analyzed by a suitable family of continuous wavelets. The analysis is therefore associated with a set of operators that describe the signal transformations at hand. These operators are then associated with a set of selectivity criteria. This leads to a set of filters that are tuned to the moving objects of interest.