In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling. For this, we consider both separable compactly-supported wavelets and boundary wavelets. We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. We also provide numerical experiments that corroborate our theoretical results. arXiv:1403.0172v2 [math.FA] number of samples as reconstruction elements. Our results extend the previous one to dimension two, although higher dimensional results can be obtained in a straightforward manner. This is an important extension, since most of the above applications involve two-or three-dimensional images. The crucial part that makes our result non-trivial is the allowance of non-diagonal scaling matrices neglecting straightforward arguments for separable two dimensional wavelets from 1D to 2D. Moreover, we will not only prove the linearity for standard twodimensional separable wavelets, but also for two-dimensional boundary wavelets which are of particular interest for smooth images. This case was not considered in [7] but was addressed recently in [1] for the case on 1D nonuniform Fourier samples. Here, for simplicity, we consider only uniform samples but in the 2D setting.At this stage we note that other higher dimensional concepts, such as curvelets and shearlets, can provide better approximations rates for cartoon-like-images; a specific class of functions ([10], [20]). However, this paper serves as an extension of known 1D results [7]. It thus provides a necessary first step in the study of reconstructions from Fourier samples within the context of generalized sampling in higher-dimensional settings. We shall discuss shearlets in an upcoming paper.Let us now make one further remark. The reader may at this stage wonder why, given a vector y of Fourier samples of a 2D image, one cannot simply form the vector x = U −1 dft y, and then form z = V dwt x and hope that z would represent wavelet coefficients of the function f to be reconstructed (here U dft and V dwt denote the discrete Fourier and wavelet transforms respectively). Unfortunately, x represents a discretization of the truncated Fourier series of f . Thus, ignoring the wavelet crime [25, p. 232] for a moment, we find that z represents the wavelet coefficients of the truncated Fourier series and not the wavelet coefficients of f itself (taking the wavelet crime into account, z would actually be an approximation to the wavelet coefficients of the truncated Fourier series). Thus, z will typically have lost all the decay properties of the original wavelet coefficients. Moreover, if we map z back to the image domain we get x = V −1 dwt z and thus we do not gain anything as x is the discretized truncated Fourier series.This paper is about getting the actual wavelet coefficients of f from the Fourier samples, thus preserving all the decay properties of the original coefficients. Thi...

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Digital contact tracing approaches based on Bluetooth low energy (BLE) have the potential to efficiently contain and delay outbreaks of infectious diseases such as the ongoing SARS-CoV-2 pandemic. In this work we propose a machine learning based approach to reliably detect subjects that have spent enough time in close proximity to be at risk of being infected. Our study is an important proof of concept that will aid the battery of epidemiological policies aiming to slow down the rapid spread of COVID-19.

In this paper, we describe a video coding design that enables a higher coding efficiency than the HEVC standard. The proposed video codec follows the design of block-based hybrid video coding, but includes a number of advanced coding tools. A part of the incorporated advanced concepts was developed by the Joint Video Exploration Team, while others are newly proposed. The key aspects of these newly proposed tools are the following. A video frame is subdivided into rectangles of variable size using a binary partitioning with variable split ratios. Three new approaches for generating spatial intra prediction signals are supported: A line-wise application of conventional intra prediction modes, coupled with a mode-dependent processing order, a region-based template matching prediction method and intra prediction modes based on neural networks. For motion-compensated prediction, a multi-hypothesis mode with more than two motion hypotheses can be used. In transform coding, mode dependent combinations of primary and secondary transforms are applied. Moreover, scalar quantization is replaced by trellis-coded quantization and the entropy coding of the quantized transform coefficients is improved. The intra and inter prediction signals can be filtered using an edge-preserving diffusion filter or a non-linear DCT-based thresholding operation.