This paper addresses the problem of estimating, analyzing and tracking objects moving with spatio-temporal rotational motion (spin or orbit). It is assumed that the digital signals of interest are acquired from a camera and structured as digital image sequences. The trajectories in the signal are two-dimensional spatial projections in time of motion taking place in a three-dimensional space. The purpose of this work is to focus on the rotational motion, i.e. estimate the angular velocity. In natural scenes, rotational motion usually composes with translational or accelerated motion on a trajectory. This paper shows that trajectory parameters and rotational motion can be efficiently estimated and tracked either simultaneously or separately. The final goal of this work is to provide selective reconstructions of moving objects of interest. This paper constructs new continuous wavelet transforms that can be tuned to both translational and rotational motion. The parameters of analysis that are taken into account in these rotational wavelet transforms are space and time position, velocity, spatial scale, angular orientation and angular velocity. The continuous wavelet functions are finally discretized for signal processing. The link between rotational motion, symmetry and critical sampling is also presented. Applications are presented with tracking and estimation.

This paper addresses the problem of estimating and analyzing accelerated motion in spatio-temporal discrete signals. It is assumed that the digital signals of interest are acquired from imaging sensors and structured as digital image sequences. The motion trajectories in the signal are two-dimensional spatial projections in time of three-dimensional motions. Consequently, they contain all the orders of acceleration. The purpose of this work is to estimate the trajectory and the motion parameters of selected moving objects in the scene. The final goal is to provide selective reconstructions of accelerated objects of interest. This paper presents the construction of new continuous wavelet transforms that can be tuned to any order of accelerations, we demonstrate their existence and provide the related admissibility conditions. The parameters for analysis that are taken into account in these accelerated wavelet transforms are spatial and temporal translations, velocity, acceleration (second or nth order), spatial scale and spatial rotation. The continuous wavelet functions are finally discretized for signal processing.

The purpose of this paper is to develop a motion based segmentation for digital image sequences that is based on the continuous wavelet transform. The continuous wavelet transform allows estimating the motion parameters on all the moving discontinuities, edges and boundaries in the image sequence. This technique provides all the information of motion parameter estimates and edge locations at once without going back and forth refining the segmentation and the motion parameter estimation. Also, this is achieved without involving any point/block corresponding techniques in our algorithm. The edges and the motion parameter estimates are calculated locally on small windows or pixels in the image planes by maximizing the square of the modulus of the wavelet transform. A clustering procedure allows separating all the detected edges into clusters of homogeneous motion. Building a ridge skeleton on the reconstructed edges in each cluster provides the ultimate motion-based segments or partition. The algorithm was simulated using real traffic image sequences acquired by a mobile camera and proved to be accurate and robust.

This paper addresses the problem of estimating, analyzing and tracking objects moving with spatio-temporal rotational motion (i.e. the angular velocity of spinning or orbiting motions). It is assumed that the digital signals of interest are acquired from a camera and structured as digital image sequences. The trajectories in the signal are two-dimensional spatial projections in time of motion taking place in a three-dimensional space. The purpose of this work is to focus on the rotational motion, i.e. estimate the angular velocity and acceleration. In natural scenes, rotational motion usually evolves on a trajectory and then composes with translational or accelerated motion. This paper shows that the trajectory parameters and the rotational parameters can be efficiently estimated and tracked either simultaneously or separately. The final goal of this work is to provide selective reconstructions of moving objects of interest. This paper constructs new continuous wavelet transforms that can be tuned to both translational and rotational motion. The parameters of analysis that are taken into account in these rotational wavelet transforms are space and time position, velocity, spatial scale, angular orientation and angular velocity. The continuous wavelet functions are finally discretized for signal processing. The link between rotational motion, symmetry and critical sampling is also presented. Simulations have been performed with estimation, detection and tracking on natural scenes.

We present a selective overview of time-frequency analysis and some of its key problems. In particular we motivate the introduction of wavelet and wavelet packet analysis. Different types of decompositions of an idealized time-frequency plane provide the basis for understanding the performance of the numerical algorithms and their corresponding interpretations within the continuous models. As examples we show how to control the frequency spreading of wavelet packets at high frequencies using nonstationary filtering and study some properties of periodic wavelet packets. Furthermore we derive a formula to compute the time localization of a wavelet packet from its indexes which is exact for linear phase filters, and show how this estimate deteriorates with deviation from linear phase.

Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context.>