Frame representations (e.g. wavelets) and subspace projections are important tools in many image processing applications. A unified framework for frames and subspace bases, as well as bases and subspace frames, is developed for finite dimensional vector spaces. Dual (subspace) bases and frames are constructed and the theory is generalized to weighted norms and seminorms. It is demonstrated how ...

Using results on syzygy modules over a multivariate polynomial ring, we are able to construct compactly supported wavelet bi-frames with few generators from almost any pair of compactly supported multivariate refinable functions. In our examples, we focus on wavelet bi-frames whose primal and dual wavelets are both derived from one box spline function. Our wavelet bi-frames have fewer generator...

‎The sequences of the form ${E_{mb}g_{n}}_{m‎, ‎ninmathbb{Z}}$,‎ ‎where $E_{mb}$ is the modulation operator‎, ‎$b>0$ and $g_{n}$ is the‎ ‎window function in $L^{2}(mathbb{R})$‎, ‎construct Fourier-like‎ ‎systems‎. ‎We try to consider some sufficient conditions on the window‎ ‎functions of Fourier-like systems‎, ‎to make a frame and find a dual‎ ‎frame with the same structure‎. ‎We also extend t...

We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with, the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also provide explicit oblique du...

This paper proves that every frame of windowed exponentials satisfies a Strong Homogeneous Approximation Property with respect to its canonical dual frame, and a Weak Homogeneous Approximation Property with respect to an arbitrary dual frame. As a consequence, a simple proof of the Nyquist density phenomenon satisfied by frames of windowed exponentials with one or finitely many generators is ob...

Frame theory is closely intertwined with signal processing by providing a canon of methodologies for the analysis of signals using (redundant) linear measurements. The dual frame associated with a frame then provides a means for reconstruction by a least squares approach. The novel paradigm of sparsity entered this area lately in various ways. Of those, in this survey paper, we will focus on th...

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L2(R). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted `2-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L...

We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with, the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also provide explicit oblique du...

finite normalized tight frames are interesting because they provide decompositionsin applications and some physical interpretations. in this article,we give a recursive method for constructing them.