Wavelet Projection Research Articles

Real-time face pose estimation

Methods were investigated for estimating the poses of human faces undergoing large rotations in depth. Dimensionality reduction using principal components analysis enabled pose changes to be visualised as manifolds in low-dimensional subspaces and provided a useful mechanism for investigating these changes. Appearance-based matching using Gabor wavelets was developed for real-time face tracking and pose estimation. A real-time Gabor wavelet projection was implemented using a Datacube MaxVideo 250 whilst an alternative system for real-time pose estimation used only standard PC hardware.

Wavelet projections of the Kuramoto-Sivashinsky equation I. Heteroclinic cycles and modulated traveling waves for short systems

Abstract We study fairly low dimensional projections of the Kuramoto-Sivashinsky partial differential equation, with periodic boundary conditions on a short interval, onto bases spanned by periodic wavelets. Such projections break the translation-reflection symmetry ( O (2)), replacing it by a finite dihedral group Dk. However, we show that, for the Perrier-Basdevant wavelets used here, the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved. In particular, we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets. We use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and Fourier modes, which preserve full ( O (2)) symmetry and are also optimal in that Fourier truncations maximise the energy (L2 norm) among all finite dimensional models. This study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains.

Approximations of Wavelet Projections

Methods for evaluating inner products [formula], of a function ƒ and a scaling function φ satisfying a dilation equation are studied. It is assumed that the values of ƒ are known at the points {2 m j} j∈ Z only.

Wavelet Projections for Radiosity

One important goal of image synthesis research is to accelerate the process of obtaining realistic images using the radiosity method. Two important concepts recently introduced are the general framework of projection methods and the hierarchical radiosity method. Wavelet theory, which explores the space of hierarchical basis functions, offers an elegant framework that unites these two concepts and allows us to more formally understand the hierarchical radiosity method. Wavelet expansions of the radiosity kernel have negligible entries in regions where high frequency/fine detail information is not needed. A sparse system remains if these entries are ignored. This is similar to applying a lossy compression scheme to the form factor matrix. The sparseness of the system allows for asymptotically faster radiosity algorithms by limiting the number of matrix terms that need to be computed. The application of these methods to 3D environments is described in4. Due to space limitations in that paper many of the subtleties of the construction could not be explored there. In this paper we discuss some of the mathematical details of wavelet projections and investigate the application of these methods to the radiosity kernel of a flatland environment, where many aspect are easier to visualize.

Coherent structures in random media and wavelets

We construct low-dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows. These models are derived from the wavelet based Galerkin projection of the PDE describing the flow and account for the (local) interaction of a small number of coherent structures. We show that, under rather general assumptions, the wavelet projection is close to the proper orthogonal decomposition, in the average energy sense. In the specific case of the 1D Kuramoto-Sivashinsky equation, we show (numerically) that the dynamics of our model agrees qualitatively with that of the original KS equation.

Comparison of spectral decomposition methods

Comparison of spectral decomposition methods