Piecewise Constant Signals Research Articles

A Markov random field model for change points detection

Detecting Change Points (CPs) in data sequences is a challenging problem that arises in a variety of disciplines, including signal processing and time series analysis. While many methods exist for PieceWise Constant (PWC) signals, relatively fewer address PieceWise Linear (PWL) signals due to the challenge of preserving sharp transitions. This paper introduces a Markov Random Field (MRF) model for detecting changes in slope. The number of CPs and their locations are unknown. The proposed method incorporates PWL prior information using MRF framework with an additional boolean variable called Line Process (LP), describing the presence or absence of CPs. The solution is then estimated in the sense of maximum a posteriori. The LP allows us to define a non-convex non-smooth energy function that is algorithmically hard to minimize. To tackle the optimization challenge, we propose an extension of the combinatorial algorithm DPS, initially designed for CP detection in PWC signals. Also, we present a shared memory implementation to enhance computational efficiency. Numerical studies show that the proposed model produces competitive results compared to the state-of-the-art methods. We further evaluate the performance of our method on three real datasets, demonstrating superior and accurate estimates of the underlying trend compared to competing methods.

Concatenated Nanopore DNA Codes.

In nanopore sequencers, single-stranded DNA molecules (or k-mers) enter a small opening in a membrane called a nanopore and modulate the ionic current through the pore, producing a channel output in the form of a noisy piecewise constant signal. An important problem in DNA-based data storage is finding a set of k-mers, i.e. a DNA code, that is robust against noisy sample duplication introduced by nanopore sequencers. Good DNA codes should contain as many k-mers as possible that produce distinguishable current signals (squiggles) as measured by the sequencer. The dissimilarity between squiggles can be estimated using a bound on their pairwise error probability, which is used as a metric for code design. Unfortunately, code construction using the union bound is limited to small k's due to the difficulty of finding maximum cliques in large graphs. In this paper, we construct large codes by concatenating codewords from a base code, thereby packing more information in a single strand while retaining the storage efficiency of the base code. To facilitate decoding, we include a circumfix in the base code to reduce the effect of the nanopore channel memory. We show that the decoding complexity scales as [Formula: see text], where m is the number of concatenated k-mers. Simulations show that the base code error rate is stable as m increases.

A robust nonlinear tracking MPC using qLPV embedding and zonotopic uncertainty propagation

In this paper, we propose a novel Nonlinear Model Predictive Control (NMPC) framework for tracking for piece-wise constant reference signals. The main novelty is the use of quasi-Linear Parameter Varying (qLPV) embeddings in order to describe the nonlinear dynamics. Furthermore, these embeddings are exploited by an extrapolation mechanism, which provides the future behaviour of the scheduling parameters with bounded estimation error. Therefore, the resulting NMPC becomes computationally efficient (comparable to a Quadratic Programming algorithm), since, at each sampling period, the predictions are linear. Benefiting from artificial target variables, the method is also able to avoid feasibility losses due to large set-point variations. Robust constraint satisfaction, closed-loop stability, and recursive feasibility certificates are provided, thanks to uncertainty propagation zonotopes and parameter-dependent terminal ingredients. A benchmark example is used to illustrate the effectiveness of the method, which is compared to state-of-the-art techniques.

On derivative-free sample-and-hold control with prescribed performance

A feedback controller is proposed to perform output reference tracking with prescribed performance for nonlinear continuous-time systems of relative degree two. The controller is of sampled-data type, i.e., measurements are available only at sampling times – a typical situation in real systems when sensors are involved. Furthermore, only output information is available, i.e., neither the full state nor derivatives can be used for feedback. A sufficient uniform sampling rate is derived and the control consists of piecewise constant signals on the sampling intervals, i.e., zero-order hold. Feasibility of the controller and satisfaction of the control objective are rigorously proven. The controller is illustrated by a numerical example.

Virtual reference feedback tuning for linear discrete-time systems with robust stability guarantees based on set membership

In this paper we propose a novel methodology that allows us to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) virtual reference feedback tuning for enforcing suitable performance and (ii) the set membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through set membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as linear matrix inequality constraints within an optimization problem whose linear cost function relies on virtual reference feedback tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop.The proposed method is not direct due to the set membership identification. However, the uncertainty set is used with the objective of providing robust stability guarantees for the closed-loop system and it is not used for the definition of the cost function, which instead is based on data. The effectiveness of the method is shown with reference to three simulation examples. A comparison with other data-driven methods is carried out.

A Sparse Recovery Algorithm Using the Neurodynamic System With Predefined Time Convergence

Compressed sensing aims to compress sparse signals losslessly into significantly smaller samples and recover them if needed. One of the most critical approaches to recovering the sparse signal is minimizing l1-norm, the convex relaxation of l0-norm with the limit of linear measurement. However, sometimes such as the recovery of piecewise constant sparse signal, the need for controlling the convergence time has become an inevitable problem to solve. To solve this problem, the predefined time pseudo-inverse locally competitive algorithm (PPLCA) based on a nonlinear neurodynamic system is proposed in this paper, which is exceptionally effective if the signal is piecewise constant. Further, under the restricted isometry property (RIP), the predefined-time convergence rate of the proposed PPLCA is proven using Lyapunov’s stability theory. Finally, the numerical simulations show that the PPLCA could converge to the stationary point within the predefined time.

Generalized multiple change-point detection in the structure of multivariate, possibly high-dimensional, data sequences

The extensive emergence of big data techniques has led to an increasing interest in the development of change-point detection algorithms that can perform well in a multivariate, possibly high-dimensional setting. In the current paper, we propose a new method for the consistent estimation of the number and location of multiple generalized change-points in multivariate, possibly high-dimensional, noisy data sequences. The number of change-points is allowed to increase with the sample size and the dimensionality of the given data sequence. Having a number of univariate signals, which constitute the unknown multivariate signal, our algorithm can deal with general structural changes; we focus on changes in the mean vector of a multivariate piecewise-constant signal, as well as changes in the linear trend of any of the univariate component signals. Our proposed algorithm, labeled Multivariate Isolate–Detect (MID) allows for consistent change-point detection in the presence of frequent changes of possibly small magnitudes in a computationally fast way.

A robust model predictive control algorithm for input–output LPV systems using parameter extrapolation

In this paper, we introduce a novel robust Model Predictive Control (MPC) algorithm for Linear Parameter Varying (LPV) systems represented in the Input–Output (IO) form. The proposed scheme embeds integral action and ensures output reference tracking for piece-wise constant signals. The algorithm is based on the online extrapolation of the LPV scheduling parameters, which are generated recursively based on a simple Taylor expansion argument. Closed-loop asymptotic stability, recursive feasibility of the online optimisation, as well as robustness towards bounded disturbances and scheduling parameter prediction uncertainties are demonstrated. Two distinct nonlinear multi-input multi-output benchmarks are used to illustrate the effectiveness of the proposed method: a numeric simulation example, used to compare the method to state-of-the-art techniques, and a high-fidelity twin rotor system, for which the method is further validated. Real-time capabilities of the proposed scheme are highlighted, since it only requires one Quadratic Program to be evaluated per discrete-time sample, during the implementation.

Supervised Learning of Analysis-Sparsity Priors With Automatic Differentiation

Sparsity priors are commonly used in denoising and image reconstruction. For analysis-type priors, a dictionary defines a representation of signals that is likely to be sparse. In most situations, this dictionary is unknown, and is to be recovered from pairs of ground-truth signals and measurements, by minimizing the reconstruction error. This defines a hierarchical optimization problem, which can be cast as a bi-level optimization. Yet, this problem is unsolvable, as reconstructions and their derivative <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> the dictionary have no closed-form expression. However, reconstructions can be iteratively computed using the Forward-Backward splitting (FB) algorithm. In this paper, we approximate reconstructions by the output of the FB algorithm, up to a linear transformation. Then, we leverage automatic differentiation to evaluate the gradient of this output <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w.r.t.</i> the dictionary, which we learn with projected gradient descent. Experiments show that our algorithm successfully learns the 1D Total Variation (TV) dictionary from piecewise constant signals. For the same case study, we propose to constrain our search to dictionaries of 0-centered columns, which removes undesired local minima and improves numerical stability.

Interpolation-based, minimum-time piecewise constant control of linear continuous-time SISO systems

We present a novel approach to the system inversion problem for linear, scalar (i.e. single-input, single-output, or SISO) plants. The problem is formulated as a constrained optimization program, whose objective function is the transition time between the initial and the final values of the system’s output, and the constraints are (i) a threshold on the input intensity and (ii) the requirement that the system’s output interpolates a given set of points. The system’s input is assumed to be a piecewise constant signal. It is formally proved that, in this frame, the input intensity is a decreasing function of the transition time. This result lets us to propose an algorithm that, by a bisection search, finds the optimal transition time for the given constraints. The algorithm is purely algebraic, and it does not require the system to be minimum phase or nonhyperbolic. It can deal with time-varying systems too, although in this case it has to be viewed as a heuristic technique, and it can be used as well in a model-free approach. Numerical simulations are reported that illustrate its performance. Finally, an application to a mobile robotics problem is presented, where, using a linearizing pre-controller, we show that the proposed approach can be applied also to nonlinear problems.

Self-triggered MPC with performance guarantee for tracking piecewise constant reference signals

This paper considers a self-triggered MPC controller design strategy for tracking piecewise constant reference signals. The proposed triggering scheme is based on the relaxed dynamic programming inequality and the idea of reference governor; such a scheme computes both the updated control action and the next triggering time. The resulting self-triggered tracking MPC control law preserves stability and constraint satisfaction and also satisfies certain a priori chosen performance requirements without the need to impose stabilizing terminal conditions. An illustrative example shows the effectiveness of this self-triggered tracking MPC implementation.

Minimizing L 1 over L 2 norms on the gradient

In this paper, we study the L 1/L 2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L 1/L 2 is better than the L 1 norm when approximating the L 0 norm to promote sparsity. Consequently, we postulate that applying L 1/L 2 on the gradient is better than the classic total variation (the L 1 norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L 1/L 2 over L 1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of magnetic resonance imaging and computed tomography reconstruction. Finally, we reveal some empirical evidence on the superiority of L 1/L 2 over L 1 when recovering piecewise constant signals from low-frequency measurements to shed light on future works.

An Augmented Lagrangian Model for Signal Segmentation

In this paper, we provide a new insight to the two-phase signal segmentation problem. We propose an augmented Lagrangian variational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.

Denoising of piecewise constant signal based on total variation

Denoising of piecewise constant signal based on total variation

Graph signal denoising using [formula omitted]-shrinkage priors

Graph signal denoising problem is studied by estimating a piecewise constant signal over an undirected graph. A new Bayesian approach is proposed that first converts a general graph to a chain graph via the depth first search algorithm, and then imposes a heavy-tailed t-shrinkage prior on the differences between consecutive signals over the induced chain graph. Posterior computation can be conveniently conducted by fully exploring the conjugacy structure in the model. Posterior contraction rate is derived for a general class of Bayesian shrinkage estimators under suitable assumptions, including the proposed t-shrinkage prior as a special case, and it is shown that this rate is optimal up to a logarithmic factor, besides automatically adapting to the unknown edge sparsity level of the graph. The excellent empirical performance of the proposed method is demonstrated via extensive simulation studies and applications to stock market data and signal denoising over large real-world networks.

Detecting multiple generalized change-points by isolating single ones

We introduce a new approach, called Isolate-Detect (ID), for the consistent estimation of the number and location of multiple generalized change-points in noisy data sequences. Examples of signal changes that ID can deal with are changes in the mean of a piecewise-constant signal and changes, continuous or not, in the linear trend. The number of change-points can increase with the sample size. Our method is based on an isolation technique, which prevents the consideration of intervals that contain more than one change-point. This isolation enhances ID’s accuracy as it allows for detection in the presence of frequent changes of possibly small magnitudes. In ID, model selection is carried out via thresholding, or an information criterion, or SDLL, or a hybrid involving the former two. The hybrid model selection leads to a general method with very good practical performance and minimal parameter choice. In the scenarios tested, ID is at least as accurate as the state-of-the-art methods; most of the times it outperforms them. ID is implemented in the R packages IDetect and breakfast, available from CRAN.

Piecewise Constant Signal Estimation by the Recording of Signal Measurements in a Linear Scheme

Piecewise Constant Signal Estimation by the Recording of Signal Measurements in a Linear Scheme

Nonconvex Energy Minimization with Unsupervised Line Process Classifier for Efficient Piecewise Constant Signals Reconstruction

In this paper, we focus on the problem of signal smoothing and step-detection for piecewise constant signals. This problem is central to several applications such as human activity analysis, speech or image analysis, and anomaly detection in genetics. We present a two-stage approach to minimize the well-known line process model which arises from the probabilistic representation of the signal and its segmentation. In the first stage, we minimize a TV least square problem to detect the majority of the continuous edges. In the second stage, we apply a combinatorial algorithm to filter all false jumps introduced by the TV solution. The performances of the proposed method were tested on several synthetic examples. In comparison to recent step-preserving denoising algorithms, the acceleration presents a superior speed and competitive step-detection quality.

A Novel Online Change Point Detection Using an Approximate Random Blanket and the Line Process Energy

In this paper, we focus on the problem of change point detection in piecewise constant signals. This problem is central to several applications such as human activity analysis, speech or image analysis and anomaly detection in genetics. We present a novel window-sliding algorithm for an online change point detection. The proposed approach considers a local blanket of a global Markov Random Field (MRF) representing the signal and its noisy observation. For each window, we define and solve the local energy minimization problem to deduce the gradient on each edge of the MRF graph. The gradient is then processed by an activation function to filter the weak features and produce the final jumps. We demonstrate the effectiveness of our method by comparing its running time and several detection metrics with state of the art algorithms.

A sparsity-promoting image decomposition model for depth recovery

This paper proposes a novel image decomposition model for scene depth recovery from low-quality depth measurements and its corresponding high resolution color image. Through our observation, the depth map mainly contains smooth regions separated by additive step discontinuities, and can be simultaneously decomposed into a local smooth surface and an approximately piecewise constant component. Therefore, the proposed unified model combines the least square polynomial approximation (for smooth surface) and a sparsity-promoting prior (for piecewise constant) to better portray the 2D depth signal intrinsically. As we know, the representation of the piecewise constant signal in gradient domain is extremely sparse. Previous researches using total variation filter based on L1-norm or Lp-norm (0 < p < 1) are both sub-optimal when addressing the tradeoff between enhancing the sparsity and keeping the model convex. We propose a novel non-convex penalty based on Moreau envelope, which promotes the prior sparsity and simultaneously maintains the convexity of the whole model for each variable. We prove the convexity of the proposed model and give the convergence analysis of the algorithm. We also introduce an iterative reweighted strategy applied on the sparsity prior to deal with the depth-color inconsistent problem and to locate the depth boundaries. Moreover, we provide an accelerated algorithm to deal with the problem of non-uniform down-sampling when transforming the depth observation matrix into the Fourier domain for fast processing. Experimental results demonstrate that the proposed method can handle various types of depth degradation and achieve promising performance in terms of recovery accuracy and running time.