An Adaptive Smoothness Parameter Strategy for Variational Optical Flow Model

The smoothness parameter is used to balance the weight of the data term and the smoothness term in variational optical flow model, which plays very significant role for the optical flow estimation, but existing methods fail to obtain the optimal smoothness parameters (OSP). In order to solve this problem, an adaptive smoothness parameter strategy is proposed. First, an amalgamated simple linear iterative cluster (SLIC) and local membership function (LMF) algorithm is used to segment the entire image into several superpixel regions. Then, image quality parameters (IQP) are calculated, respectively, for each superpixel region. Finally, a neural network model is applied to compute the smoothness parameter by these image quality parameters of each superpixel region. Experiments were done in three public datasets (Middlebury, MPI_Sintel, and KITTI) and our self-constructed outdoor dataset with the proposed method and other existing classical methods; the results show that our OSP method achieves higher accuracy than other smoothness parameter selection methods in all these four datasets. Combined with the dual fractional order variational optical flow model (DFOVOFM), the proposed model shows better performance than other models in scenes with illumination inhomogeneity and abnormity. The OSP method fills the blank of the research of adaptive smoothness parameter, pushing the development of the variational optical flow models.

Recovery of continuous functions of two variables from their Fourier coefficients known with error

In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions. As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions. We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]

Epsilon Best Arm Identification in Spectral Bandits

We propose an analysis of Probably Approximately Correct (PAC) identification of an ϵ-best arm in graph bandit models with Gaussian distributions. We consider finite but potentially very large bandit models where the set of arms is endowed with a graph structure, and we assume that the arms' expectations μ are smooth with respect to this graph. Our goal is to identify an arm whose expectation is at most ϵ below the largest of all means. We focus on the fixed-confidence setting: given a risk parameter δ, we consider sequential strategies that yield an ϵ-optimal arm with probability at least 1-δ. All such strategies use at least T*(μ)log(1/δ) samples, where R is the smoothness parameter. We identify the complexity term T*(μ) as the solution of a min-max problem for which we give a game-theoretic analysis and an approximation procedure. This procedure is the key element required by the asymptotically optimal Track-and-Stop strategy.

SC-MEB: spatial clustering with hidden Markov random field using empirical Bayes

Spatial transcriptomics (ST) has been emerging as a powerful technique for resolving gene expression profiles while retaining the tissue spatial information. These spatially resolved transcriptomics make it feasible to examine the complex mulitcellular systems under different microenvironments. To answer scientific questions with spatial transcriptomics, the first step is to identify cell clusters by integrating available spatial information which would expand the understanding of how cell types and states are regulated by tissues. Here, we introduce SC-MEB, an empirical Bayes approach for spatial clustering analysis using hidden Markov random field. We also derive an efficient expectation-maximization (EM) algorithm based on iterative conditional mode (ICM) for SC-MEB. Compared with BayesSpace, a recently developed method, SC-MEB is not only computationally efficient and scalable to the sample size increment, but also is capable of choosing the smoothness parameter and the number of clusters as well. We then performed comprehensive simulation studies to demonstrate the effectiveness of SC-MEB over some existing methods. We first applied SC-MEB to perform clustering analysis in the spatial transcriptomics dataset from human dorsolateral prefrontal cortex tissues that were manually annotated. Our analysis results show that SC-MEB can achieve similar clustering performance with BayesSpace that uses the true number of clusters and fixed smoothness parameter. We then applied SC-MEB to analyze the dataset from a patient with colorectal cancer (CRC) and COVID-19, and further performed differential expression analysis to identify signature genes related to the clustering results. The heatmap for identified signature genes shows that the identified clusters from SC-MEB is more separable than those from BayesSpace. By using pathway analysis, we identified three immune-related clusters and further compared mean expressions of COVID-19 signature genes in immune regions with those in non-immune ones. As such, SC-MEB provides a valuable computational tool for investigating structural organizations of tissues from spatial transcriptomic data.

On fractional Orlicz–Sobolev spaces

Some recent results on the theory of fractional Orlicz–Sobolev spaces are surveyed. They concern Sobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, and criteria for compact embeddings. The limits of these spaces when the smoothness parameter $$s\in (0,1)$$ s ∈ ( 0 , 1 ) tends to either of the endpoints of its range are also discussed. This note is based on recent papers of ours, where additional material and proofs can be found.

Spatially Variable Advection Correction of Doppler Radial Velocity Data

Techniques to mitigate analysis errors arising from the nonsimultaneity of data collections typically use advection-correction procedures based on the hypothesis (frozen turbulence) that the analyzed field can be represented as a pattern of unchanging form in horizontal translation. It is more difficult to advection correct the radial velocity than the reflectivity because even if the vector velocity field satisfies this hypothesis, its radial component does not—but that component does satisfy a second-derivative condition. We treat the advection correction of the radial velocity (υr) as a variational problem in which errors in that second-derivative condition are minimized subject to smoothness constraints on spatially variable pattern-translation components (U, V). The Euler–Lagrange equations are derived, and an iterative trajectory-based solution is developed in which U, V, and υr are analyzed together. The analysis code is first verified using analytical data, and then tested using Atmospheric Imaging Radar (AIR) data from a band of heavy rainfall on 4 September 2018 near El Reno, Oklahoma, and a decaying tornado on 27 May 2015 near Canadian, Texas. In both cases, the analyzed υr field has smaller root-mean-square errors and larger correlation coefficients than in analyses based on persistence, linear time interpolation, or advection correction using constant U and V. As some experimentation is needed to obtain appropriate parameter values, the procedure is more suitable for non-real-time applications than use in an operational setting. In particular, the degree of spatial variability in U and V, and the associated errors in the analyzed υr field are strongly dependent on a smoothness parameter.

Scattering of a Klein–Gordon particle by a smooth barrier

We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scattering solutions are given in terms of the Whittaker Mκ,μ(x) function. The reflection and transmission coefficients are calculated in terms of the energy, the height, and the smoothness of the potential barrier. For any value of the smoothness parameter we observed transmission resonances.

Reliability Analysis of Layered Soil Slopes Considering Different Spatial Autocorrelation Structures

It is widely recognized that different geological formations often vary differently in space. Therefore, soil properties from different layers should be modeled by different autocorrelation functions (ACFs) to reflect such soil heterogeneity. However, the same ACFs are frequently used for different soil layers in slope reliability analysis for simplicity purpose in the literature. The present work is a study on the effects of ACFs on the reliability analysis of layered soil slopes, where the soil properties of different layers are considered by different ACFs. Five commonly used classical ACFs and the non-classical Whittle–Matérn model were investigated in this study. Cholesky decomposition and Monte Carlo simulation were used to simulate the spatial variability of the soil properties and estimate the probability of failure (Pf) of slopes, respectively. Illustrative examples with various parametric studies show that when the soil properties from different layers are characterized by the same ACFs, the Pf of the studied slopes is comparable with that estimated using different ACFs for different soil layers. This indicates that the type of ACF has only a small impact on the slope reliability assessment. However, the Pf may be underestimated by the single exponential ACF and overestimated by the cosine exponential ACF. The scale of fluctuation of the soil properties influences the slope reliability more than the ACFs. In addition, the smoothness parameter in the non-classical model has a significant influence on the reliability of the slope, where Pf increases with the increase of the smoothness parameter.

An Automatic Baseline Correction Method Based on the Penalized Least Squares Method

Baseline drift spectra are used for quantitative and qualitative analysis, which can easily lead to inaccurate or even wrong results. Although there are several baseline correction methods based on penalized least squares, they all have one or more parameters that must be optimized by users. For this purpose, an automatic baseline correction method based on penalized least squares is proposed in this paper. The algorithm first linearly expands the ends of the spectrum signal, and a Gaussian peak is added to the expanded range. Then, the whole spectrum is corrected by the adaptive smoothness parameter penalized least squares (asPLS) method, that is, by turning the smoothing parameter λ of asPLS to obtain a different root-mean-square error (RMSE) in the extended range, the optimal λ is selected with minimal RMSE. Finally, the baseline of the original signal is well estimated by asPLS with the optimal λ. The paper concludes with the experimental results on the simulated spectra and measured infrared spectra, demonstrating that the proposed method can automatically deal with different types of baseline drift.