Spatial and temporal regularization to estimate COVID-19 Reproduction Number R(t): Promoting piecewise smoothness via convex optimization

Among the different indicators that quantify the spread of an epidemic, such as the on-going COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the monitoring of the evolution of this number, a new estimation procedure is proposed here, assuming a well-accepted model for current incidence data, based on past observations. The novelty of the proposed approach is twofold: 1) the estimation of the reproduction number is achieved by convex optimization within a proximal-based inverse problem formulation, with constraints aimed at promoting piecewise smoothness; 2) the approach is developed in a multivariate setting, allowing for the simultaneous handling of multiple time series attached to different geographical regions, together with a spatial (graph-based) regularization of their evolutions in time. The effective-ness of the approach is first supported by simulations, and two main applications to real COVID-19 data are then discussed. The first one refers to the comparative evolution of the reproduction number for a number of countries, while the second one focuses on French counties and their joint analysis, leading to dynamic maps revealing the temporal co-evolution of their reproduction numbers.

Cell-filtering-based multi-scale Shannon–Cosine wavelet denoising method for locust slice images

The locust slice images have all the features such as strong self-similarity, piecewise smoothness and nonlinear texture structure. Multi-scale interpolation operator is an effective tool to describe such structures, but it cannot overcome the influence of noise on images. Therefore, this research designed the Shannon–Cosine wavelet which possesses all the excellent properties such as interpolation, smoothness, compact support and normalization, then constructing multi-scale wavelet interpolative operator, the operator can be applied to decompose and reconstruct the images adaptively. Combining the operator with the local filter operator (mean and median), a multi-scale Shannon–Cosine wavelet denoising algorithm based on cell filtering is constructed in this research. The algorithm overcomes the disadvantages of multi-scale interpolation wavelet, which is only suitable for describing smooth signals, and realizes multi-scale noise reduction of locust slice images. The experimental results show that the proposed method can keep all kinds of texture structures in the slice image of locust. In the experiments, the locust slice images with mixture noise of Gaussian and salt–pepper are taken as examples to compare the performances of the proposed method and other typical denoising methods. The experimental results show that the Peak Signal-To-Noise Ratio (PSNR) of the denoised images obtained by the proposed method is greater 27.3%, 24.6%, 2.94%, 22.9% than Weiner filter, wavelet transform method, median and average filtering, respectively; and the Structural Similarity Index (SSIM) for measuring image quality is greater 31.1%, 31.3%, 15.5%, 10.2% than other four methods, respectively. As the variance of Gaussian white noise increases from 0.02 to 0.1, the values of PSNR and SSIM obtained by the proposed method only decrease by 11.94% and 13.33%, respectively, which are much less than other 4 methods. This shows that the proposed method possesses stronger adaptability.

Hyperspectral Feature Extraction Using Sparse and Smooth Low-Rank Analysis

In this paper, we develop a hyperspectral feature extraction method called sparse and smooth low-rank analysis (SSLRA). First, we propose a new low-rank model for hyperspectral images (HSIs) where we decompose the HSI into smooth and sparse components. Then, these components are simultaneously estimated using a nonconvex constrained penalized cost function (CPCF). The proposed CPCF exploits total variation penalty, ℓ 1 penalty, and an orthogonality constraint. The total variation penalty is used to promote piecewise smoothness, and, therefore, it extracts spatial (local neighborhood) information. The ℓ 1 penalty encourages sparse and spatial structures. Additionally, we show that this new type of decomposition improves the classification of the HSIs. In the experiments, SSLRA was applied on the Houston (urban) and the Trento (rural) datasets. The extracted features were used as an input into a classifier (either support vector machines (SVM) or random forest (RF)) to produce the final classification map. The results confirm improvement in classification accuracy compared to the state-of-the-art feature extraction approaches.

Application of spatially related MRF model in NMF hyperspectral unmixing

Aiming at Non-negative Matrix Factorization (NMF)’s problem of initialization and "local minima" in hyperspectral unmixing, a NMF linear unmixing algorithm with spatial correlation constrains (SCNMF) based on Markov Random Field (MRF) was proposed. Firstly, Hyperspectral Signal identification by minimum error (HySime) method was adopted to estimate the number of endmembers, initialized endmember matrix and abundance matrix by Vertex Component Analysis (VCA) and Fully Constrained Least Squares (FCLS) respectively. then established energy function to depict the spatial distribution characteristics of ground objects by MRF model. Finally, spatial correlation constraint based on MRF model and NMF standard objective function were combined in the form of altemating iteration to estimate endmember spectrum and abundance of hyperspectral image. Theoretical analysis and experimental results indicated that, the endmember decomposition precision of SCNMF is 10.6% higher than that of Minimum Volume Constrained NMF (MVC-NMF), 12.3% higher than that of Piecewise Smoothness NMF with Sparseness Constraints(PSNMFSC), 14.1% higher than that of NMF with Alternating Projected Subgradients(APS-NMF); the abundance decomposition precision of SCNMF is 14.4% higher than that of MVC-NMF, 15.9% higher than that of PSNMFSC, 15.3% higher than that of APS-NMF.The proposed SCNMF can remedy NMF's deficiency in describing spatial correlation characteristics, and decrease spatial energy distribution error.

ON THE PIECEWISE SMOOTHNESS OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS FOR A LARGER CLASS OF INITIAL DATA

We prove that if the initial data do not belong to a certain subset of Ck, then the solutions of scalar conservation laws are piecewise Cksmooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a Ck+1shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of 𝒮(ℝ) of the first category. It should be pointed out that this subset is smaller than those used in previous works. We point out that Thom's theory of catastrophes, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.