Abstract: We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339–360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute $k$th order discrete derivatives over the input points. Perhaps not surprisingly, trend filtering estimates appear to have the structure of $k$th degree spline functions, with adaptively chosen knot points (we say “appear” here as trend filtering estimates are not really functions over continuous domains, and are only defined over the discrete set of inputs). This brings to mind comparisons to other nonparametric regression tools that also produce adaptive splines; in particular, we compare trend filtering to smoothing splines, which penalize the sum of squared derivatives across input points, and to locally adaptive regression splines [Ann. Statist. 25 (1997) 387–413], which penalize the total variation of the $k$th derivative. Empirically, we discover that trend filtering estimates adapt to the local level of smoothness much better than smoothing splines, and further, they exhibit a remarkable similarity to locally adaptive regression splines. We also provide theoretical support for these empirical findings; most notably, we prove that (with the right choice of tuning parameter) the trend filtering estimate converges to the true underlying function at the minimax rate for functions whose $k$th derivative is of bounded variation. This is done via an asymptotic pairing of trend filtering and locally adaptive regression splines, which have already been shown to converge at the minimax rate [Ann. Statist. 25 (1997) 387–413]. At the core of this argument is a new result tying together the fitted values of two lasso problems that share the same outcome vector, but have different predictor matrices.

Abstract: A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete H2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L2 norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014

Abstract: This paper presents a system to reconstruct piecewise planar and compact floorplans from images, which are then converted to high quality texture-mapped models for free- viewpoint visualization. There are two main challenges in image-based floorplan reconstruction. The first is the lack of 3D information that can be extracted from images by Structure from Motion and Multi-View Stereo, as indoor scenes abound with non-diffuse and homogeneous surfaces plus clutter. The second challenge is the need of a sophisti- cated regularization technique that enforces piecewise pla- narity, to suppress clutter and yield high quality texture mapped models. Our technical contributions are twofold. First, we propose a novel structure classification technique to classify each pixel to three regions (floor, ceiling, and wall), which provide 3D cues even from a single image. Second, we cast floorplan reconstruction as a shortest path problem on a specially crafted graph, which enables us to enforce piecewise planarity. Besides producing compact piecewise planar models, this formulation allows us to di- rectly control the number of vertices (i.e., density) of the output mesh. We evaluate our system on real indoor scenes, and show that our texture mapped mesh models provide compelling free-viewpoint visualization experiences, when compared against the state-of-the-art and ground truth.

Abstract: We present a new algorithm for computational caustic design. Our algorithm solves for the shape of a transparent object such that the refracted light paints a desired caustic image on a receiver screen. We introduce an optimal transport formulation to establish a correspondence between the input geometry and the unknown target shape. A subsequent 3D optimization based on an adaptive discretization scheme then finds the target surface from the correspondence map. Our approach supports piecewise smooth surfaces and non-bijective mappings, which eliminates a number of shortcomings of previous methods. This leads to a significantly richer space of caustic images, including smooth transitions, singularities of infinite light density, and completely black areas. We demonstrate the effectiveness of our approach with several simulated and fabricated examples.

Abstract: Scene flow is defined as the motion field in 3D space, and can be computed from a single view when using an RGBD sensor. We propose a new scene flow approach that exploits the local and piecewise rigidity of real world scenes. By modeling the motion as a field of twists, our method encourages piecewise smooth solutions of rigid body motions. We give a general formulation to solve for local and global rigid motions by jointly using intensity and depth data. In order to deal efficiently with a moving camera, we model the motion as a rigid component plus a non-rigid residual and propose an alternating solver. The evaluation demonstrates that the proposed method achieves the best results in the most commonly used scene flow benchmark. Through additional experiments we indicate the general applicability of our approach in a variety of different scenarios.

Abstract: In this paper, two new techniques for microwave imaging of layered structures are introduced. These techniques were developed to address the limiting issues associated with classical synthetic aperture radar (SAR) imaging techniques in generating focused and properly-positioned images of embedded objects in generally layered dielectric structures. The first method, referred to as piecewise SAR (PW-SAR), is a natural extension of the classical SAR technique, and considers physical and electrical properties of each individual layer and the discontinuity among them. Although this method works well with low loss dielectric media, its applicability to lossy media is limited. This is due to the fact that this method does not consider signal attenuation. Moreover, multiple reflections within each layer are not incorporated. To improve imaging performance in which these important phenomena are included, a second method was developed that utilizes the Green's function of the layered structure and casts the imaging approach into a deconvolution procedure. Subsequently, a Wiener filter-based deconvolution technique is used to solve the problem. The technique is referred to as Wiener filter-based layered SAR (WL-SAR). The performance and efficacy of these SAR based imaging techniques are demonstrated using simulations and corresponding measurements of several different layered media.

Abstract: In the paper, we present an efficient method to solve the piecewise constant Mumford-Shah (M-S) model for two-phase image segmentation within the level set framework. A clustering algorithm is used to find approximately the intensity means of foreground and background in the image, and so the M-S functional is reduced to the functional of a single variable (level set function), which avoids using complicated alternating optimization to minimize the reduced M-S functional. Experimental results demonstrated some advantages of the proposed method over the well-known Chan-Vese method using alternating optimization, such as robustness to the locations of initial contour and the high computation efficiency.

Abstract: In this paper, we propose a variational Bayesian method for Retinex to simulate and interpret how the human visual system perceives color. To construct a hierarchical Bayesian model, we use the Gibbs distributions as prior distributions for the reflectance and the illumination, and the gamma distributions for the model parameters. By assuming that the reflection function is piecewise continuous and illumination function is spatially smooth, we define the energy functions in the Gibbs distributions as a total variation function and a smooth function for the reflectance and the illumination, respectively. We then apply the variational Bayes approximation to obtain the approximation of the posterior distribution of unknowns so that the unknown images and hyperparameters are estimated simultaneously. Experimental results demonstrate the efficiency of the proposed method for providing competitive performance without additional information about the unknown parameters, and when prior information is added the proposed method outperforms the non-Bayesian-based Retinex methods we compared.

Abstract: The orbital boundary value problem, also known as Lambert Problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an efficient initial guess to an Householder iterative method that is then able to converge, for the single revoltuion case, in only two iterations. The resulting algorithm is compared to Gooding's procedure revealing to be numerically as accurate, while having a smaller computational complexity.

Abstract: We study and extend the recently introduced total generalized variation (TGV) functional for multichannel images. This functional has already been established to constitute a well-suited convex model for piecewise smooth scalar images. It comprises exactly the functions of bounded variation but is, unlike purely total-variation based functionals, also aware of higher-order smoothness. For the multichannel version which is developed in this paper, basic properties and existence of minimizers for associated variational problems regularized with second-order TGV is shown. Furthermore, we address the design of numerical solution methods for the minimization of functionals with TGV\(^2\) penalty and present, in particular, a class of primal-dual algorithms. Finally, the concrete realization for various image processing problems, such as image denoising, deblurring, zooming, dequantization and compressive imaging, are discussed and numerical experiments are presented.

Abstract: A fast pattern matching scheme termed matching by tone mapping (MTM) is introduced which allows matching under nonlinear tone mappings. We show that, when tone mapping is approximated by a piecewise constant/linear function, a fast computational scheme is possible requiring computational time similar to the fast implementation of normalized cross correlation (NCC). In fact, the MTM measure can be viewed as a generalization of the NCC for nonlinear mappings and actually reduces to NCC when mappings are restricted to be linear. We empirically show that the MTM is highly discriminative and robust to noise with comparable performance capability to that of the well performing mutual information, but on par with NCC in terms of computation time.

Abstract: We propose a fast splitting approach to the classical variational formulation of the image partitioning problem, which is frequently referred to as the Potts or piecewise constant Mumford--Shah model. For vector-valued images, our approach is significantly faster than the methods based on graph cuts and convex relaxations of the Potts model which are presently the state-of-the-art. The computational costs of our algorithm only grow linearly with the dimension of the data space which contrasts the exponential growth of the state-of-the-art methods. This allows us to process images with high-dimensional codomains such as multispectral images. Our approach produces results of a quality comparable with that of graph cuts and the convex relaxation strategies, and we do not need an a priori discretization of the label space. Furthermore, the number of partitions has almost no influence on the computational costs, which makes our algorithm also suitable for the reconstruction of piecewise constant (color or vect...

Abstract: In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

Abstract: A piecewise analytical function is proposed and applied to investigate the steady-state behavior of series-parallel resonant converter operated in a discontinuous current mode. The converter shows two sequences of the equivalent circuits alternatively operated in the discontinuous current mode with different output voltages. To get the response time of current in one resonance, a successive solving process based on the state-space method is presented analytically in each sequence. This solving process can describe the complicated behavior resulting from the load rectifier, which makes the output capacitor appear or disappear several times in one switching period. By introducing the output voltage coefficient and the principle of energy transmission balance, the steady-state model is deduced afterward. This model is accurate and simple, making it helpful to design and optimize the converter conveniently. An excellent agreement is obtained when comparing numerical values calculated by the proposed model to the simulation and to the experimental results.

Abstract: An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features.

Abstract: Differential equations with piecewise constant argument (EPCA) were proposed for investigations in [63, 91] by founders of the theory, K. Cook, S. Busenberg, J. Wiener, and S. Shah. They are named as differential EPCA. In the last three decades, many interesting results have been obtained, and applications have been realized in this theory. Existence and uniqueness of solutions, oscillations and stability, integral manifolds and periodic solutions, and many other questions of the theory have been intensively discussed. Besides the mathematical analysis, various models in biology, mechanics, and electronics were developed by using these systems. The founders proposed that the method of investigation of these equations is based on a reduction to discrete systems. That is, only values of solutions at moments, which are integers or multiples of integers, were discussed. Moreover, systems must be linear with respect to the values of solutions, if the argument is not deviated. It reduces the theoretical depth of the investigations as well as the number of real-world problems, which can be modeled by using these equations.

Abstract: State-of-the-art Multi-View Stereo (MVS) algorithms deliver dense depth maps or complex meshes with very high detail, and redundancy over regular surfaces. In turn, our interest lies in an approximate, but light-weight method that is better to consider for large-scale applications, such as urban scene reconstruction from ground-based images. We present a novel approach for producing dense reconstructions from multiple images and from the underlying sparse Structure-from-Motion (SfM) data in an efficient way. To overcome the problem of SfM sparsity and textureless areas, we assume piecewise planarity of man-made scenes and exploit both sparse visibility and a fast over-segmentation of the images. Reconstruction is formulated as an energy-driven, multi-view plane assignment problem, which we solve jointly over superpixels from all views while avoiding expensive photoconsistency computations. The resulting planar primitives--defined by detailed superpixel boundaries--are computed in about 10 seconds per image.

Abstract: Aim The application of island biogeography theory in habitat fragmentation research assumes a simple relationship between species richness and fragment area. However, previous work has suggested that in some cases thresholds can be detected, at which the form of the island species–area relationship (ISAR) changes abruptly. Piecewise regression has been advocated as a suitable statistical technique to model such thresholds. Here we first provide a comparative analysis of piecewise regression models to determine the prevalence and type of thresholds in habitat island ISARs. Second, we evaluate piecewise regression as a method for locating thresholds in the ISAR, with particular emphasis on the implications of data transformation. Location World-wide. Methods Seventy-six habitat island datasets were sourced from the literature. An information theoretic approach was employed to compare linear regression ISAR models with piecewise regression models. The models were applied to untransformed (species–area), semi-log (species–log area) and log–log (log species–log area) data. Three types of piecewise regression models were evaluated: continuous, discontinuous and zero slope. Model performance was compared using the Akaike information criterion. We also examined the influence on model performance of taxon, number of habitat islands, and area of smallest island. Results Linear regression models performed best, although piecewise models were preferred in a number of cases. Cases in which no model was significant were most prevalent in untransformed space relative to the semi-log and log–log transformations. Piecewise fits were more prevalent in datasets with a larger numbers of islands. Main conclusions Data transformation is a key part of model selection and needs to be explicitly considered, especially in terms of drawing inferences from models. Piecewise models, even if selected as the favoured model in our analyses, were often ecologically unintelligible in relation to area alone. When detected, breakpoint values ranged over five orders of magnitude, although with one exception all were under 50 ha. Our findings highlight the limitations of using individual threshold values to inform conservation practice.

Abstract: Multi-task feature learning has been proposed to improve the generalization performance by learning the shared features among multiple related tasks and it has been successfully applied to many real-world problems in machine learning, data mining, computer vision and bioinformatics. Most existing multi-task feature learning models simply assume a common noise level for all tasks, which may not be the case in real applications. Recently, a Calibrated Multivariate Regression (CMR) model has been proposed, which calibrates different tasks with respect to their noise levels and achieves superior prediction performance over the non-calibrated one. A major challenge is how to solve the CMR model efficiently as it is formulated as a composite optimization problem consisting of two non-smooth terms. In this paper, we propose a variant of the calibrated multi-task feature learning formulation by including a squared norm regularizer. We show that the dual problem of the proposed formulation is a smooth optimization problem with a piecewise sphere constraint. The simplicity of the dual problem enables us to develop fast dual optimization algorithms with low per-iteration cost. We also provide a detailed convergence analysis for the proposed dual optimization algorithm. Empirical studies demonstrate that, the dual optimization algorithm quickly converges and it is much more efficient than the primal optimization algorithm. Moreover, the calibrated multi-task feature learning algorithms with and without the squared norm regularizer achieve similar prediction performance and both outperform the non-calibrated ones. Thus, the proposed variant not only enables us to develop fast optimization algorithms, but also keeps the superior prediction performance of the calibrated multi-task feature learning over the non-calibrated one.

Abstract: We first develop a spectrally accurate Petrov--Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively trea...

Abstract: This paper presents a method for the 3D reconstruction of a piecewise-planar surface from range images, typically laser scans with millions of points. The reconstructed surface is a watertight polygonal mesh that conforms to observations at a given scale in the visible planar parts of the scene, and that is plausible in hidden parts. We formulate surface reconstruction as a discrete optimization problem based on detected and hypothesized planes. One of our major contributions, besides a treatment of data anisotropy and novel surface hypotheses, is a regularization of the reconstructed surface w.r.t. the length of edges and the number of corners. Compared to classical area-based regularization, it better captures surface complexity and is therefore better suited for man-made environments, such as buildings. To handle the underlying higher-order potentials, that are problematic for MRF optimizers, we formulate minimization as a sparse mixed-integer linear programming problem and obtain an approximate solution using a simple relaxation. Experiments show that it is fast and reaches near-optimal solutions.

Abstract: This paper presents an approximate method for general fractional order dynamic systems. Firstly, a novel piecewise approximate method is proposed for fractional order integrator based on its frequency distributed mode. Based on the above method, an integer order approximation system is constructed to approximate a fractional order system. Theoretical analysis results show that the proposed method can achieve much better performance than the existing schemes for a given order in an interested frequency range. The advantage of the proposed method lies in that the resulting system are standard integer order system, which facilitates systematic stability analysis and controller synthesis in view of the well-developed linear or nonlinear system theory. Numerical simulations are presented to illustrate the effectiveness of the proposed approach in the end.

Abstract: Switched linear hyperbolic partial differential equation sa re considered in this technical note. They model infinite dimensional systems of conservation laws and balance laws, which are potentially affecte db y a distributed source or sink term. The dynamics and the boundary condi- tions are subject to abrupt changes given by a switching signal, modeled as a piecewise constant function and possibly a dwell time. By means of Lyapunov techniques some sufficient conditions are obtained for the expo- nential stability of the switching system, uniformly for all switching signals. Different cases are considered with or without a dwell time assumption on the switching signals, and on the number of positive characteristic veloci- ties (which may also depend on the switching signal). Some numerical sim- ulations are also given to illustrate some main results, and to motivate this study.

Abstract: In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an individual sequence manner. We use a tree notion in order to partition the space of regressors in a nested structure. The introduced algorithms adapt not only their regression functions but also the complete tree structure while achieving the performance of the “best” linear mixture of a doubly exponential number of partitions, with a computational complexity only polynomial in the number of nodes of the tree. While constructing these algorithms, we also avoid using any artificial “weighting” of models (with highly data dependent parameters) and, instead, directly minimize the final regression error, which is the ultimate performance goal. The introduced methods are generic such that they can readily incorporate different tree construction methods such as random trees in their framework and can use different regressor or partitioning functions as demonstrated in the paper.