We describe a procedure that enables us to construct dual pairs of wavelet frames from certain dual pairs of Gabor frames. Applying the construction to Gabor frames generated by appropriate exponential Bsplines gives wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. There is also a reverse transform, whi...

(1) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark. (2) Department of Mathematical Sciences, KAIST, Daejeon, Korea. (3) Department of Mathematics, Yeungnam University, Gyeongsan-si,Korea. [email protected], [email protected], [email protected] This work was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by th...

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames...

The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setti...

Gabor system plays an important role in signal processing, image processing and other applications because of their redundancy properties. In this paper, symmetric or antisymmetric Gabor frames in two dimensions about any symmetric points are constructed from Gabor frames given. In special, symmetric or antisymmetric Parseval Gabor frames with better properties are obtained. Some thoughts of ex...

In this paper we construct frames of Gabor type for the space Lrad(R ) of radial L-functions, and more generally, for subspaces of modulation spaces consisting of radial distributions. Hereby, each frame element itself is a radial function. This construction is based on a generalization of the so called Feichtinger-Gröchenig theory – sometimes also called coorbit space theory – which was develo...

This paper examines the perception of first- and second-order motion in human vision. In an extension of previous work by Boulton and Baker [J.B. Boulton, C.L. Baker, Motion detection is dependent on spatial frequency not size, Vision Res., 31 (1991) 77-87; J.B. Boulton, C.L. Baker, Different parameters control motion perception above and below a critical density, Vision Res., 33 (1993) 1803-18...

Digital Gabor filters are indispensable tools of local time-frequency analysis in signal processing. With strong orientation selectivity, discrete (tight) Gabor frames generated by 2D Gabor filters also see their wide applications in image processing and volume data processing. However, owing to the lack of multi-scale structures, discrete Gabor frames are less effective than multiresolution an...

Recently, several researchers have proposed using spatio-temporal filters for image motion analysis. For example, the optical flow field can be calculated from the output of a set of spatio-temporal filters. Some of the most popular spatio-temporal filters are the space-time Gabor filters, obtained by convolving a time varying image with a space-time Gabor function. Based on the cellular neural...

Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two communit...