Piecewise Constant Structure Research Articles

A General Non-Lipschitz Infimal Convolution Regularized Model: Lower Bound Theory and Algorithm

The convex infimal convolution model proposed in Chambolle and Lions [Numer. Math., 76 (1997), pp. 167--188] is a fundamental model to extract two useful components from a single input image and has various low level vision applications. In many of them, one target component has an (approximately) piecewise constant structure and the other is a smoothly varying function or repeated texture pattern. In this paper, we propose and study a general non-Lipschitz infimal convolution (GnLIC) regularization model, which covers most existing applications in this type. Therein the non-Lipschitz regularization enforces the piecewise constant property of the first component. For this GnLIC model, we prove a lower bound theory for its local minimizers and a local version for its stationary points. Motivated by these, we naturally extend previous works to design an inexact iterative support shrinking algorithm with proximal linearization for our GnLIC model (InISSAPL-GnLIC). Moreover, we establish the sequence convergence property and a sequence lower bound theory for InISSAPL-GnLIC, provided that an inexact subgradient condition generated by a subsolver holds. The subsolver is constructed by efficient ADMM and a specially designed feasibilization operation. We finally give numerical experiments in two low level vision applications: Retinex and cartoon-texture decomposition. These tests demonstrate that our non-Lipschitz regularization based method can indeed extract the piecewise constant component better than existing approaches, which is consistent with the established lower bound theory.

Image retinex based on the nonconvex TV-type regularization

<p style='text-indent:20px;'>Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity of the image based on the nonconvex <inline-formula><tex-math id="M1">\begin{document}$ \ell^p $\end{document}</tex-math></inline-formula>-quasinorm were limited. To deal with this problem, this paper considers a TV<inline-formula><tex-math id="M2">\begin{document}$ _p $\end{document}</tex-math></inline-formula>-HOTV<inline-formula><tex-math id="M3">\begin{document}$ _q $\end{document}</tex-math></inline-formula>-based retinex model with <inline-formula><tex-math id="M4">\begin{document}$ p, q\in(0, 1) $\end{document}</tex-math></inline-formula>. Specially, the TV<inline-formula><tex-math id="M5">\begin{document}$ _p $\end{document}</tex-math></inline-formula> term based on the total variation(TV) regularization can describe the reflectance efficiently, which has the piecewise constant structure. The HOTV<inline-formula><tex-math id="M6">\begin{document}$ _q $\end{document}</tex-math></inline-formula> term based on the high order total variation(HOTV) regularization can penalize the smooth structure called the illumination. Since the proposed model is non-convex, non-smooth and non-Lipschitz, we employ the iteratively reweighed <inline-formula><tex-math id="M7">\begin{document}$ \ell_1 $\end{document}</tex-math></inline-formula> (IRL1) algorithm to solve it. We also discuss some properties of our proposed model and algorithm. Experimental experiments on the simulated and real images illustrate the effectiveness and the robustness of our proposed model both visually and quantitatively by compared with some related state-of-the-art variational models.</p>

A convex analysis approach to multi-material topology optimization

This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available materials. A “multi-bang” framework based on convex analysis is proposed where the desired piecewise constant structure is incorporated using a convex penalty term. Together with a suitable tracking term, this allows formulating the problem of optimizing the topology of the distribution of material parameters as minimizing a convex functional subject to a (nonlinear) equality constraint. The applicability of this approach is validated for two model problems where the control enters as a potential and a diffusion coefficient, respectively. This is illustrated in both cases by numerical results based on a semi-smooth Newton method.

Input Delay Compensation in Multi-Agent Systems via Piecewise Constant Control

We consider a multi-agent system with time-invariant topology containing several agents governed by similar linear time-invariant ODEs with constant delay in actuation. Piecewise constant (PWC) control with sampling time larger than the delay value is applied. Assuming the delay-free version of the system can be brought to consensus by the PWC feedback, we design a predictive PWC feedback achieving consensus in the system with delayed input. During prediction, PWC structure is employed to determine future input to the system from its past output. This input is then used to predict the future state and compensate for the delay.

Unsupervised Segmentation of Spectral Images with a Spatialized Gaussian Mixture Model and Model Selection

In this article, we describe a novel unsupervised spectral image segmentation algorithm. This algorithm extends the classical Gaussian Mixture Model-based unsupervised classification technique by incorporating a spatial flavor into the model: the spectra are modelized by a mixture of K classes, each with a Gaussian distribution, whose mixing proportions depend on the position. Using a piecewise constant structure for those mixing proportions, we are able to construct a penalized maximum likelihood procedure that estimates the optimal partition as well as all the other parameters, including the number of classes. We provide a theoretical guarantee for this estimation, even when the generating model is not within the tested set, and describe an efficient implementation. Finally, we conduct some numerical experiments of unsupervised segmentation from a real dataset.

Optimal LQ-Type Switched Control Design for a Class of Linear Systems with Piecewise Constant Inputs

This paper is devoted to a specific LQ-type optimal control problem (OCP) in the presence of additional control constraints. We consider control processes governed by linear differential equations with a priori known control switchings. The piecewise constant structure of the admissible controls under consideration is motivated by variety of concrete engineering applications and moreover, can be interpreted as a result of a quantization procedure applied to the original dynamics. We propose an new implementable numeric algorithm that makes it possible to calculate a consistent approximating solution to the initial constrained LQ-type OCPs. The contribution discusses some theoretic aspects of the obtained computational scheme and also contains a numerical example.

Hartog’s Type Extension Theorem in Some Piecewise Constant Structure Relations For Metamonogenic Functions of First Order

The main purpose of this paper is to study a Hartog’s type extension theorem for a more general class of functions, called, metamonogenic and multi-metamonogenic functions, with a Clifford type algebraic structure, called, some piecewise constant structure relations.

A Cauchy–Pompeiu Representation Formula Using Dirac Operator and its Applications in Some Piecewise Constant Structure Relations

In the present paper we use the piecewise constant structure relations of a Clifford algebra in order to obtain a Cauchy–Pompeiu representation for D − λ and D + λ operators, with these formulas we construct a distributional solutions for the equations that involves these operators with arbitrary right hand side. We also present an example where we build an integral representation for combinations of these operators.

Recovery of localised structure from signals with non-sparse components

We consider the recovery of localised structure from signals consisting of a piecewise constant structure and sparse components. A new algorithm is presented which aims to reconstruct signals of this type from a limited set of observed data. The algorithm is broken down into two subproblems which both involve minimisation of an $l_1$-regularised least squares problem. Numerical results are presented which demonstrate the effectiveness and efficiency of the proposed method. References M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo. A Fast Algorithm for the Constrained Formulation of Compressive Image Reconstruction and Other Linear Inverse Problems. IEEE International Conference on Acoustics Speech and Signal Processing, pages 4034--4037, 2010. doi:10.1109/ICASSP.2010.5495758 R. L. Broughton, I. D. Coope, P. F. Renaud, and R. E. H. Tappenden. A Box Constrained Gradient Projection Algorithm for Compressed Sensing. Signal Processing, 91(8):1985--1992, August 2011. doi:10.1016/j.sigpro.2011.03.003 E. Candes, J. Romberg, and T. Tao. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Transactions on Information Theory, 52(2):489--509, 2004. doi:10.1109/TIT.2005.862083 E. J. Candes. Compressive Sampling. In Proceedings of the International Congress of Mathematics, pages 1433--1452, Madrid, Spain, 2006. European Mathematical Society. E. J. Candes and T. Tao. Near-Optimal Signal Recovery From Random Projections and Universal Encoding Strategies. IEEE Transactions on Information Theory, 52(12):5406--5425, December 2006. doi:10.1109/TIT.2006.885507 D. L. Donoho. Compressed Sensing. IEEE Transactions on Information Theory, 52(4):1289--1306, 2006. doi:10.1109/TIT.2006.871582 M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE Journal of Selected Topics in Signal Processing, 1(4):586--597, December 2007. doi:10.1109/JSTSP.2007.910281 G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 3 edition, 1996. A. Griffin and P. Tsakalides. Compressed sensing of audio signals using multiple sensors. Lausanne, Switzerland, 25--29 August 2008. EUSIPCO08. P. C. Hansen. {Rank-Deficient and Discrete Ill-Posed Problems}. SIAM, 1998. M. Lustig, D. Donoho, and J. Pauly. Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magnetic Resonance in Imaging, 58(6):1182--1195, December 2007. doi:10.1002/mrm.21391 J. Provost and F. Lesage. The Application fo Compressed Sensing for Photo-Acoustic Tomography. IEEE Transactions on Medical Imaging, 28(4), April 2009. doi:10.1109/TMI.2008.2007825 J. L. Starck and J. Bobin. Astronomical Data Analysis and Sparsity: From Wavelets to Compressed Sensing. Proceedings of the IEEE, 98(6):1021--1030, 2010. doi:10.1109/JPROC.2009.2025663 H. Yu and G. Wang. Compressed sensing based interior tomography. Physics in Medicine and Biology, pages 2791--2805, 2009. doi:10.1088/0031-9155/54/9/014

General Algebraic Structures of Clifford Type and Cauchy-Pompeiu Formulae for Some Piecewise Constant Structure Relations

The paper starts with generalization of Clifford algebras using other structure relations which possibly depend on spacelike variables. For piecewise constant structure relations the paper constructs fundamental solutions explicitly and proves a Cauchy-Pompeiu Integral Formula.

Extracting Default Probabilities from Sovereign Bonds

Sovereign risk analysis is central in debt markets. Considering different bonds and countries, there are numerous measures aiming to identify the way risk is perceived by market participants. In such environment, probabilities of default play a central role in investors’ decisions. This article contributes by providing a parametric arbitrage-free dynamic model to estimate defaultable term structures of sovereign bonds. The proposed model builds on Duffie and Singleton’s (1999) general reduced-form model by proposing a piecewise constant structure for the conditional probabilities of defaults. Once an average recovery rate value is fixed for the whole market, the proposed model estimates implied probabilities of defaults from bond prices, working as a parsimonious tool to quantify investor’s perception of credit risk. We apply this methodology to analyze the behavior of default probabilities within the Brazilian sovereign fixed income market at three different recent economic moments.

Computation of time-optimal controls using a second-variation descent search

A second-variation descent search is proposed for computing time-optimal control functions for linear time-invariant plants. Since the piecewise constant structure of the optimal control function parameterizes the relevant two-point boundary value problem, the terms required in the variational analysis are simply first and second partial derivatives of an appropriate scalar cost function with respect to the parameters. Closed form expressions for these derivatives can be obtained. As a result, the second-order terms are incorporated into the analysis with only a modest increase in computational effort over that required for the first-variation methods appearing in the literature [1]-[4]. The usual second-variation algorithm can be applied only if the matrix of second derivatives is positive definite at each iteration. Since this condition generally fails to hold during the early stages of the search, a modified version of the algorithm is developed, based on the selection of a positive definite model for the matrix of second derivatives. Performance of the modified algorithm is shown to be significantly better than that of the familiar Newton algorithm in regions of parameter space far from the solution point. Particularly striking results are obtained when all the eigenvalues of the plant are real.