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.

Anisotropic Diffusion Combined with Nonconvex Functional for Noise Image Segmentation

Aiming at the problems of low segmentation accuracy of noise image, poor noise immunity of the existing models and poor adaptability to complex noise environment, a noise image segmentation algorithm using anisotropic diffusion and nonconvex functional was proposed. First, focusing on the “staircase effect”, a nonconvex functional was introduced into the energy functional model for smooth denoising. Second, the validity and accuracy of the model were established by proving that there was no global minimum in the solution space of the nonconvex energy functional model; the improved model was then used to obtain a smooth and clear image edge while maintaining the edge integrity. Third, the smooth image obtained from the nonconvex energy functional model was combined with the level set model to obtain the anisotropic diffusion gray level set model. The optimal outline of the target was obtained by calculating the minimum value of the energy functional. Finally, an anisotropic diffusion equation with nonconvex energy functional model was built in this algorithm to segment noise image accurately and quickly. A series of comparative experiments on the proposed algorithm and similar algorithms were conducted. The results showed that the proposed algorithm had strong noise resistance and provided precise segmentation for noise image.

Convergence analysis of time-discretisation schemes for rate-independent systems

It is well known that rate-independent systems involving nonconvex energy functionals in general do not allow for time-continuous solutions even if the given data are smooth. In the last years, several solution concepts were proposed that include discontinuities in the notion of solution, among them the class of global energetic solutions and the class of BV-solutions. In general, these solution concepts are not equivalent and numerical schemes are needed that reliably approximate that type of solutions one is interested in. In this paper, we analyse the convergence of solutions of three time-discretisation schemes, namely an approach based on local minimisation, a relaxed version of it and an alternate minimisation scheme. For all three cases, we show that under suitable conditions on the discretisation parameters discrete solutions converge to limit functions that belong to the class of BV-solutions. The proofs rely on a reparametrisation argument. We illustrate the different schemes with a toy example.

Linearly constrained evolutions of critical points and an application to cohesive fractures

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite-dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite-dimensional. Nevertheless, in the infinite-dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite-dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one- and two-dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

The Restoration of Textured Images Using Fractional-Order Regularization

Image restoration problem is ill-posed, so most image restoration algorithms exploit sparse prior in gradient domain to regularize it to yield high-quality results, reconstructing an image with piecewise smooth characteristics. While sparse gradient prior has good performance in noise removal and edge preservation, it also tends to remove midfrequency component such as texture. In this paper, we introduce the sparse prior in fractional-order gradient domain as texture-preserving strategy to restore textured images degraded by blur and/or noise. And we solve the unknown variables in the proposed model using method based on half-quadratic splitting by minimizing the nonconvex energy functional. Numerical experiments show our algorithm's robust outperformance.

Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

In this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.