New approaches to operator-theoretic frameworks for wavelet multiresolution approximation and analysis

The shortcomings of the classical Fourier transform based techniques of analyzing signals and systems are well known. The wavelet transform has evolved over the last decade as an efficient alternative to the Fourier transform because of its many advantages, the most important of which is its multiresolution analysis property. Fundamentally, a wavelet transform allows us to decompose a function f(·) ∈ L2R as f˙ n=-infinityinfinityf ˙,DmTny ˙ DmTny˙ 0.1 where Df(·) := 2 f(2(·)) and Tf(·) := f((·) - 1) The operator D is the dilation operator and T is the unit time-shift operator. {lcub}DmTnpsi(·){rcub} is the set of dilated and translated versions of the mother-wavelet psi and provide an orthonormal basis for the function space L2R .; In this work, we choose to take an operator theoretic route to understanding the wavelet transform and, ultimately, the core theory of multiresolution analysis. We use a framework in which signals are analyzed in terms of their projections onto wavelet subspaces and modify it to computationally simpler forms. In the process, we show that the wavelet transform can be computed by applying the operators D and T on either the mother-wavelet or on the signal under analysis. By reducing the distinction between a signal and the analyzing wavelets, we can optimize computation by making smart decisions about where to apply D or T for a given signal segment. We also show the unitary equivalence of D and T and explore the development of a framework where only a time-shift operator is used in a wavelet transform; this, and the instant appeal of a shift-operator, lead us to defining a time-varying shift-operator.; Towards the end of this work, we develop a variation of the traditional multiresolution analysis by defining a scale transform. This leads to a simple multiresolution-like model where the concept of detail subspaces is replaced by that of error subspaces and where we do away with almost all projections of the signal onto wavelet-subspaces. However, in gaining simplicity, we lose the important property of orthonormality of the detail subspaces. We address this issue by developing a combined framework by looking at interactions between the new transform and the traditional detail subspaces.