Lifting Framework Research Articles

On Some Structural Properties of Sampled-Data Systems

The study of some structural properties, namely controllability and spectrum distribution, in a hybrid system composed of a continuous-time system controlled by a digital controller is carried out in the lifting framework. The novelty of our approach lies in the infinite-dimensional nature of the state-space in the lifted domain due to the fact that the system evolves over a temporal continuum as opposed to the finite-dimensional state-space with infinite-dimensional input/output systems currently used in the literature. Different concepts of controllability related directly to the continuous-time behavior of sampled-data systems are introduced, namely exact, approximate and null controllability. It is shown that the two former notions never occur in sampled-data systems while the latter notion is a generic property for these systems. Some facts regarding the fundamental input/output structure of sampled-data systems from the control viewpoint are drawn from these results. The spectrum distribution of sampled-data feedback systems is also characterized.

On the relationship between the overlapping rounding transform and lifting frameworks for reversible subband transforms

A new framework for reversible subband transforms based on the overlapping rounding transform (ORT) has been proposed as an alternative to the lifting framework. In this article, we show that the ORT framework is, in fact, a special case of the lifting framework with only trivial extensions.

Nonlinear wavelet transforms for image coding via lifting

We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels. Preliminary compression results for model and real-world images demonstrate the promise of our techniques.

A multiresolution framework for variational subdivision

Subdivision is a powerful paradigm for the generaton of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a griven set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consquently, we how how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.