In the multi-window spline-type spaces, the fast computation of the realizable dual frame could be achieved through a constructive reformulation of the biorthogonal relations. In this paper, we extend the results obtained in spline-type spaces, for the constructive realization of an approximate dual Gabor-like frame. We demonstrate the advantages of this approach in both flexibility and speed. The method allows in a natural way to handle non standard Gabor constructions like nonuniformity in frequency and the reductions of the number of used modulations. Experimental tests are presented in support of the algorithm.

In this paper, we explore the possibility of applying the anisotropic generalized Hough transform (AGHT) enhanced with a Gabor based time-frequency filtering (GTF) for the determination of the mandibular canal in digital dental panoramic radiographs. The proposed method is based on template matching using the fact that the shape of the mandibular canal is usually the same, followed by a filtering of the accumulator space in the Gabor domain for a precise detection of the position. The proposed procedure consists of a detailed description of the shape of the canal in its canonical form and on preserving Gabor filtering information for sorting the hierarchy of location candidates after applying anisotropically the extraction algorithm. The experimental results show that the proposed procedure is robust to recognition under occlusion and under the presence of additional structures e.g. teeth, projection errors.

In this paper, the stability of translation-invariant spaces of distributions over locally compact groups is stated as boundedness of synthesis and projection operators. At first, a characterization of the stability of spline-type spaces is given, in the standard sense of the stability for shift-invariant spaces, that is, linear independence characterizes lower boundedness of the synthesis operator in Banach spaces of distributions. The constructive nature of the proof for Theorem 2 enabled us to constructively realize the biorthogonal system of a given one. Then, inspired by the multiresolution analysis and the Lax equivalence for general discretization schemes, we approached the stability of a sequence of spline-type spaces as uniform boundedness of projection operators. Through Theorem 3, we characterize stable sequences of stable spline-type spaces.

In this work, we consider new computational aspects to improve the approximation of Hilbert-Schmidt operators via generalized Gabor multipliers. One aspect is to consider the approximation of the symbol of an Hilbert-Schmidt operator as projection in the spline-type space associated to a Gabor multiplier. This gives the possibility to employ a selection procedure of the analysis and synthesis function, interpreted as time-frequency lag; hence, with the related algorithm it is possible to handle both underspread and overspread operators. In the numerical section, we exploit the case of approximating overspread operators having compact and smooth spreading function and discontinuous time-varying systems. For the latter, the approximation of discontinuities in the symbol is not straightforward achievable in the generalized Gabor multipliers setting. For this reason, another aspect is to further process the symbol through a Hough transform, to detect discontinuities and to smooth them using a new class of approximants. This procedure creates a bridge between features detections techniques and harmonic analysis methods and in specific cases it almost doubles the accuracy of approximation.

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