Abstract: It is often a difficult task to accurately segment images with intensity inhomogeneity, because most of representative algorithms are region-based that depend on intensity homogeneity of the interested object. In this paper, we present a novel level set method for image segmentation in the presence of intensity inhomogeneity. The inhomogeneous objects are modeled as Gaussian distributions of different means and variances in which a sliding window is used to map the original image into another domain, where the intensity distribution of each object is still Gaussian but better separated. The means of the Gaussian distributions in the transformed domain can be adaptively estimated by multiplying a bias field with the original signal within the window. A maximum likelihood energy functional is then defined on the whole image region, which combines the bias field, the level set function, and the piecewise constant function approximating the true image signal. The proposed level set method can be directly applied to simultaneous segmentation and bias correction for 3 and 7T magnetic resonance images. Extensive evaluation on synthetic and real-images demonstrate the superiority of the proposed method over other representative algorithms.

Abstract: Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on $\varepsilon$) require $\Omega(\text{poly}(1/\varepsilon))$ neurons while deep networks (i.e., networks whose depth grows with $1/\varepsilon$) require $\mathcal{O}(\text{polylog}(1/\varepsilon))$ neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU.

Abstract: The purpose of this chapter is to present a unified theory for the numerical implementation of modal methods for the analysis of electromagnetic phenomena with specific boundary conditions. All the fundamental concepts that form the basis of our study will be detailed. In plasmonics and photonics in general, solving Maxwell equations involving irregular functions is common. For example, when the relative permittivity is a piecewise constant function describing a dielectric–metal interface, the eigenmodes of the propagation equation are solutions of Maxwell's equations subject to specific boundary conditions at the interfaces between homogenous media. Prior knowledge about the eigenmodes allows one to define more appropriate expansion functions, and the rate of convergence of the numerical scheme will depend on the choice of these functions. In this chapter, we present and explain, a unified numerical formalism that allows one to build, from a set of subsectional functions defined on a set of subintervals, expansion functions defined on a global domain by enforcing certain stresses deduced from electromagnetic field properties. Then numerical modal analysis of a plasmonic device, such as a ring resonator, is presented as an example of an application.

Abstract: We present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a pri...

Abstract: Under mild conditions, the Pareto front (Pareto set) of a continuous $\boldsymbol m$ -objective optimization problem forms an ( $\boldsymbol {m-1}$ )-dimensional piecewise continuous manifold. Based on this property, this paper proposes a self-organizing multiobjective evolutionary algorithm. At each generation, a self-organizing mapping method with ( $\boldsymbol {m-1}$ ) latent variables is applied to establish the neighborhood relationship among current solutions. A solution is only allowed to mate with its neighboring solutions to generate a new solution. To reduce the computational overhead, the self-organizing training step and the evolution step are conducted in an alternative manner. In other words, the self-organizing training is performed only one single step at each generation. The proposed algorithm has been applied to a number of test instances and compared with some state-of-the-art multiobjective evolutionary methods. The results have demonstrated its advantages over other approaches.

Abstract: In order to overcome the disadvantages of logistic map in designing chaos-based cipher, the piecewise logistic map (PLM) is presented. Some properties related to cryptography of the PLM, such as ergodicity, Lyapunov exponent, and bifurcation, are analyzed and compared with the logistic map. From the view of cryptography, the PLM owns better properties than the logistic map. Then, a novel pseudorandom number generator (PRNG) based on the PLM is proposed. Since the cryptographic properties of the PLM are enhanced, the presented PRNG achieves a trade-off between efficiency and security. Both performance analysis and simulation test confirm that our scheme is simple, secure, and efficient, with high potential to be adopted as a stream cipher for secure communication.

Abstract: The high penetration of photovoltaic (PV) generators leads to a voltage rise in the distribution network. To comply with grid standards, distribution system operators need to limit this voltage rise. Reactive power control is one of the most proposed remedies. A popular form of reactive power control is an active power dependent characteristic to define the reactive power control of a PV generator. This standard Q ( P ) characteristic is a simple curve, which is not adapted to the specific situation in the grid. Therefore, this work proposes a method to define the optimal Q ( P ) curve. The optimal Q ( P ) curve is represented as a piecewise constant or a piecewise linear function. The parameters are optimized based on historical smart meter information, to obtain a Q ( P ) curve that keeps the voltage within limits throughout the whole year, with a minimal amount of reactive power. An easy to solve convex optimization problem defines the parameters. The method is applied to unbalanced three-phase four-wire systems. Several simulations with realistic data are performed on an existing distribution network to compare the optimal Q ( P ) curve with standard Q ( P ) and Q ( V ) curves.

Abstract: We present a fast, practical method for personalizing a hand shape basis to an individual user's detailed hand shape using only a small set of depth images. To achieve this, we minimize an energy based on a sum of render-and-compare cost functions called the golden energy. However, this energy is only piecewise continuous, due to pixels crossing occlusion boundaries, and is therefore not obviously amenable to efficient gradient-based optimization. A key insight is that the energy is the combination of a smooth low-frequency function with a high-frequency, low-amplitude, piecewisecontinuous function. A central finite difference approximation with a suitable step size can therefore jump over the discontinuities to obtain a good approximation to the energy's low-frequency behavior, allowing efficient gradient-based optimization. Experimental results quantitatively demonstrate for the first time that detailed personalized models improve the accuracy of hand tracking and achieve competitive results in both tracking and model registration.

Abstract: Summary We propose a likelihood ratio scan method for estimating multiple change points in piecewise stationary processes. Using scan statistics reduces the computationally infeasible global multiple-change-point estimation problem to a number of single-change-point detection problems in various local windows. The computation can be efficiently performed with order O{npt log (n)}. Consistency for the estimated numbers and locations of the change points are established. Moreover, a procedure is developed for constructing confidence intervals for each of the change points. Simulation experiments and real data analysis are conducted to illustrate the efficiency of the likelihood ratio scan method.

Abstract: We present and analyze an enriched Galerkin finite element method (EG) to solve elliptic and parabolic equations with jump coefficients. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions which can be considered as a penalty stabilization. The method is shown to be locally and globally conservative, while keeping fewer degrees of freedom in comparison with discontinuous Galerkin finite element methods (DG). Moreover, we present and analyze a fast effective EG solver whose cost is roughly that of CG and which can handle an arbitrary order of approximations. A number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of EG when coupled with transport.

Abstract: We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.

Abstract: Inverse imaging problems are inherently under-determined, and hence it is important to employ appropriate image priors for regularization. One recent popular prior---the graph Laplacian regularizer---assumes that the target pixel patch is smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. To address this problem, in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization for image denoising. Specifically, we first show the convergence of the graph Laplacian regularizer to a continuous-domain functional, integrating a norm measured in a locally adaptive metric space. Focusing on image denoising, we derive an optimal metric space assuming non-local self-similarity of pixel patches, leading to an optimal graph Laplacian regularizer for denoising in the discrete domain. We then interpret graph Laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify our analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods such as BM3D for natural images, and outperforms them significantly for piecewise smooth images.

Abstract: In the commonly employed regularization models of image restoration and super-resolution (SR), the norm determination is often challenging. This paper proposes a method to adaptively determine the optimal norms for both fidelity term and regularization term in the (SR) restoration model. Inspired by a generalized likelihood ratio test, a piecewise function is proposed to solve the norm of the fidelity term. This function can find the stable norm value in a certain number of iterations, regardless of whether the noise type is Gaussian, impulse, or mixed. For the regularization norm, the main advantage of the proposed method is that it is locally adaptive. Specifically, it assigns different norms for different pixel locations, according to the local activity measured by a structure tensor metric. The proposed method was tested using different types of images. The experimental results and error analyses verify the efficacy of the method.

Abstract: We consider the problem of predicting an outcome variable using p covariates that are measured on n independent observations, in a setting in which additive, flexible, and interpretable fits are desired. We propose the fused lasso additive model (FLAM), in which each additive function is estimated to be piecewise constant with a small number of adaptively-chosen knots. FLAM is the solution to a convex optimization problem, for which a simple algorithm with guaranteed convergence to a global optimum is provided. FLAM is shown to be consistent in high dimensions, and an unbiased estimator of its degrees of freedom is proposed. We evaluate the performance of FLAM in a simulation study and on two data sets. Supplemental materials are available online, and the R package flam is available on CRAN.

Abstract: Recently there have been exciting developments in Monte Carlo methods, with the development of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has led to some fundamentally new Monte Carlo algorithms which can be used to sample from, say, a posterior distribution. Interestingly, continuous-time algorithms seem particularly well suited to Bayesian analysis in big-data settings as they need only access a small sub-set of data points at each iteration, and yet are still guaranteed to target the true posterior distribution. Whilst continuous-time MCMC and SMC methods have been developed independently we show here that they are related by the fact that both involve simulating a piecewise deterministic Markov process. Furthermore we show that the methods developed to date are just specific cases of a potentially much wider class of continuous-time Monte Carlo algorithms. We give an informal introduction to piecewise deterministic Markov processes, covering the aspects relevant to these new Monte Carlo algorithms, with a view to making the development of new continuous-time Monte Carlo more accessible. We focus on how and why sub-sampling ideas can be used with these algorithms, and aim to give insight into how these new algorithms can be implemented, and what are some of the issues that affect their efficiency.

Abstract: In visible light communication (VLC) systems, optical orthogonal frequency-division multiplexing (O-OFDM) is an appealing modulation scheme. Recently, a number of O-OFDM schemes have been proposed. Among them, direct-current O-OFDM (DCO-OFDM) is a widely used scheme for its high spectral efficiency and low complexity. Since VLC involves a combination of illumination and communication, different optical power is often required to achieve a certain illumination level. However, clipping noise will dominate a severe performance degradation of DCO-OFDM when a relatively high or low illumination level is imposed, and it restricts applications of DCO-OFDM in future VLC systems. To address this problem, an enhanced DCO-OFDM (eDCO-OFDM) scheme is proposed. By introducing a piecewise function with adaptive slopes according to the required optical power, the proposed eDCO-OFDM scheme has the potential to effectively eliminate the clipping noise. Furthermore, two parameter selection mechanisms with different complexities and performance gains are designed for the piecewise function in the eDCO-OFDM scheme. Simulation results verify the energy and spectral efficiency of the proposed scheme under the constraints of optical power.

Abstract: An effective forward collision warning (FCW) system must be compatible with drivers' risk perceptions and behavioral responses. The Collision Avoidance Metrics Partnership (CAMP) developed a kinematic-based FCW algorithm to determine the minimum distance needed to stop safely under various levels of rear-end crash risk. The algorithm generates a linear function for predicting drivers' expected response decelerations (ERDs) by considering motions of the involved vehicles. This linear function works well when the risks perceived by drivers are low; however, at elevated risks when the lead vehicle (LV) decelerates at an unexpectedly high rate, or at high relative speeds, the warnings are triggered too late for the subject vehicle to avoid a rear-end collision. The current study extends the CAMP FCW algorithm to improve the handling of extreme high-collision-risk scenarios. A total of 111 brake-only noncollision events was presented in the Tongji University Driving Simulator, and drivers' braking behaviors were used to model their ERDs. We found that ERDs depended on the interaction of LV deceleration and relative speed. In response to this finding, a nonlinear function with an interaction term was combined with a linear function into a piecewise function that accommodated both higher and lower LV deceleration conditions. The applicable domain of the warning onset range was then computed for a wide range of kinematic conditions. Results showed the piecewise function to be a better predictor of ERD than the linear function, as well as to result in fewer driver rejections of the FCWs.

Abstract: Block-based image or video coding standards (e.g. JPEG) compress an image lossily by quantizing transform coefficients of non-overlapping pixel blocks. If the chosen quantization parameters (QP) are large, then hard decoding of a compressed image—using indexed quantization bin centers as reconstructed transform coefficients—can lead to unpleasant blocking artifacts. Leveraging on recent advances in graph signal processing (GSP), we propose a dequantization scheme specifically for piecewise smooth (PWS) images: images with sharp object boundaries and smooth interior surfaces. We first mathematically define a PWS image as a low-frequency signal with respect to an inter-pixel similarity graph with edges of weights 1 or 0. Using quantization bin boundaries as constraints, we then jointly optimize the desired graph-signal and the similarity graph in a unified framework. A generalization to consider generalized piecewise smooth (GPWS) images—where sharp object boundaries are replaced by transition regions—is also proposed. Experimental results show that our proposed scheme outperforms a state-of-the-art dequantization method by 1 dB on average in PSNR.

Abstract: The paper is focused on an application of sequential three-way decisions and granular computing to the problem of multi-class statistical recognition of the objects, which can be represented as a sequence of independent homogeneous (regular) segments. As the segmentation algorithms usually make it possible to choose the degree of homogeneity of the features in a segment, we propose to associate each object with a set of such piecewise regular representations (granules). The coarse-grained granules stand for a low number of weakly homogeneous segments. On the contrary, a sequence with a large count of high-homogeneous small segments is considered as a fine-grained granule. During recognition, the sequential analysis of each granularity level is performed. The next level with the finer granularity is processed, only if the decision at the current level is unreliable. The conventional Chow's rule is used for a non-commitment option. The decision on each granularity level is proposed to be also sequential. The probabilistic rough set of the distance of objects from different classes at each level is created. If the distance between the query object and the next checked reference object is included in the negative region (i.e., it is less than a fixed threshold), the search procedure is terminated. Experimental results in face recognition with the Essex dataset and the state-of-the-art HOG features are presented. It is demonstrated, that the proposed approach can increase the recognition performance in 2.5-6.5 times, in comparison with the conventional PHOG (pyramid HOG) method.

Abstract: Recently, there have been many progresses for the problem of non-rigid structure reconstruction based on 2D trajectories, but it is still challenging to deal with complex deformations or restricted view ranges. Promising alternatives are the piecewise reconstruction approaches, which divide trajectories into several local parts and stitch their individual reconstructions to produce an entire 3D structure. These methods show the state-of-the-art performance, however, most of them are specialized for relatively smooth surfaces and some are quite complicated. Meanwhile, it has been reported numerously in the field of pattern recognition that obtaining consensus from many weak hypotheses can give a strong, powerful result. Inspired by these reports, in this paper, we push the concept of part-based reconstruction to the limit: Instead of considering the parts as explicitly-divided local patches, we draw a large number of small random trajectory sets. From their individual reconstructions, we pull out a statistic of each 3D point to retrieve a strong reconstruction, of which the procedure can be expressed as a sparse l1-norm minimization problem. In order to resolve the reflection ambiguity between weak (and possibly bad) reconstructions, we propose a novel optimization framework which only involves a single eigenvalue decomposition. The proposed method can be applied to any type of data and outperforms the existing methods for the benchmark sequences, even though it is composed of a few, simple steps. Furthermore, it is easily parallelizable, which is another advantage.

Abstract: This paper is concerned with the dissipativity analysis and design of discrete Markovian jumping neural networks with sector-bounded nonlinear activation functions and time-varying delays represented by Takagi–Sugeno fuzzy model. The augmented fuzzy neural networks with Markovian jumps are first constructed based on estimator of Luenberger observer type. Then, applying piecewise Lyapunov–Krasovskii functional approach and stochastic analysis technique, a sufficient condition is provided to guarantee that the augmented fuzzy jump neural networks are stochastically dissipative. Moreover, a less conservative criterion is established to solve the dissipative state estimation problem by using matrix decomposition approach. Furthermore, to reduce the computational complexity of the algorithm, a dissipative estimator is designed to ensure stochastic dissipativity of the error fuzzy jump neural networks. As a special case, we have also considered the mixed $H_{\infty }$ and passive analysis of fuzzy jump neural networks. All criteria can be formulated in terms of linear matrix inequalities. Finally, two examples are given to show the effectiveness and potential of the new design techniques.

Abstract: A key element for the global optimization of non-convex mixed-integer bilinear problems is the computation of a tight lower bound for the objective function being minimized. Multiparametric disaggregation is a technique for generating a mixed-integer linear relaxation of a bilinear problem that works by discretizing the domain of one of the variables in every bilinear term according to a numeric representation system. This can be done up to a certain accuracy level that can be different for each discretized variable so as to adjust the number of significant digits to their range of values and give all variables the same importance. We now propose a normalized formulation (NMDT) that achieves the same goal using a common setting for all variables, which is equivalent to the number of uniform partitions in a closely related, piecewise McCormick (PCM) approach. Through the solution of several benchmark problems from the literature involving four distinct problem classes, we show that the computational performance of NMDT is already better than PCM for ten partitions, with the difference rising quickly due to the logarithmic versus linear growth in the number of binary variables with the number of partitions. The results also show that a global optimization solver based on the proposed relaxation compares favorably with commercial solvers BARON and GloMIQO.

Abstract: This paper develops a topology optimization procedure for three-dimensional electromagnetic waves with an edge element-based finite-element method. In contrast to the two-dimensional case, three-dimensional electromagnetic waves must include an additional divergence-free condition for the field variables. The edge element-based finite-element method is used to both discretize the wave equations and enforce the divergence-free condition. For wave propagation described in terms of the magnetic field in the widely used class of non-magnetic materials, the divergence-free condition is imposed on the magnetic field. This naturally leads to a nodal topology optimization method. When wave propagation is described using the electric field, the divergence-free condition must be imposed on the electric displacement. In this case, the material in the design domain is assumed to be piecewise homogeneous to impose the divergence-free condition on the electric field. This results in an element-wise topology optimization algorithm. The topology optimization problems are regularized using a Helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method. In order to ensure the applicability of the filter in the element-wise topology optimization version, a regularization method is presented to project the nodal into an element-wise physical density variable.