We consider equidistant discrete splines S(j), j ∈ Z, which may grow as O(|j|s) as |j| → ∞. Such splines present a relevant tool for digital signal processing. The Zak transforms of Bsplines yield the integral representation of discrete splines. We define the wavelet space as a weak orthogonal complement of the coarse-grid space in the fine-grid space. We establish the integral representation o...

In this correspondence, we generalize the theory of compactly supported biorthogonal two-channel wavelet bases to M -channel. A sufficient condition for the M -channel perfect reconstruction filter banks to constructM -channel biorthogonal bases of compactly supported wavelets is derived. It is shown that the construction of biorthogonal M -channel wavelet bases is equivalent to the design of a...

We show that a multi-dimensional scaling function of order γ (possibly fractional) can always be represented as the convolution of a polyharmonic B-spline of order γ and a distribution with a bounded Fourier transform which has neither order nor smoothness. The presence of the B-spline convolution factor explains all key wavelet properties: order of approximation, reproduction of polynomials, v...

Image-matched nonseparable wavelets can find potential use in many applications including image classification, segmentation, compressive sensing, etc. This paper proposes a novel design methodology that utilizes convolutional neural network (CNN) to design two-channel non-separable wavelet matched to a given image. The design is proposed on quincunx lattice. The loss function of the convolutio...

The selection of filter bank in wavelet compression is crucial, affecting image quality and system design. Recently, the biorthogonal coiflet (cooklet) family of wavelet filters has been constructed [2] [4], and explicit frequency domain formulae have been developed [2] in the Bernstein polynomial basis. In this paper we use the Bernstein basis for frequency domain design and construction of bi...

The use of a complex wavelet transform with similar properties to the DT-CWT in seismic processing was suggested in Fernandes (2001). A wavelet basis was used for linear inversion with application to well logging in Li et. al., (1996), using a fully decimated wavelet transform and the Haar wavelet. However, the transform used does not have the shift invariance and directional selectivity proper...

Numerical algorithms using wavelet bases are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in the new system of coordinates. As in all transform methods, such approach seeks an advantage in that the computation is faster in the new system of coordinates than in the original domain. However, due to the recursive definit...

Near-best and best wavelet packet transforms were compared with wavelet transforms for the compression of polysomnograms in a low bitrate telemonitoring application. Near-best wavelet packet transforms provided better overall performance with respect to efficiency of computation and compression, and amenability for use in more sophisticated quality-on-demand algorithms. Keywords— near-best base...

Background: Spatially resolved transcriptomics (ST) measures gene expression along with the spatial coordinates of measurements. The analysis ST data involves significant computation complexity. In this work, we propose a dimensionality reduction algorithm that retains structure. Methods: We combine wavelet transformation matrix factorization to select spatially-varying ge...