Selective diffusion involving reaction for binarization of bleed-through document images

Binarization has always been a challenging problem in document image processing because of various types of degradation. In this paper, we present a nonlinear reaction–diffusion model for binarization of bleed-through document images, which is the Perona–Malik equation involving diffusion coefficient based on structure tensor along with a nonlinear reaction term. The Perona–Malik diffusion is utilized to selectively smooth document images with bleed-through removal. Meanwhile, the nonlinear reaction term takes the responsibility for the desired binarization. In order to solve our model numerically, we develop a parallel–series splitting algorithm by combining finite differencing with two kinds of splitting methods in the literature. Our algorithm is tested on seven publicly available datasets (DIBCO 2009 to 2014 and 2016). The experimental results show that our method averagely outperforms six relevant models for the nineteen document images with bleed-through in the DIBCO series datasets.     

Finite element approximation of a forward and backward anisotropic diffusion model in image denoising and form generalization

A new forward–backward anisotropic diffusion model is introduced. The two limit cases are the Perona‐Malik equation and the Total Variation flow model. A fully discrete finite element scheme is studied using C⁰‐piecewise linear elements in space and the backward Euler difference scheme in time. A priori estimates are proven. Numerical results in image denoising and form generalization are presented.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On Lipschitz solutions for some forward–backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward–backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona–Malik model in image processing and the other that of Höllig's model related to the Clausius–Duhem inequality in the second law of thermodynamics.     

A backward–forward regularization of the Perona–Malik equation

It is shown that the Perona–Malik equation (PME) admits a natural regularization by forward–backward diffusions possessing better analytical properties than PME itself. Well-posedness of the regularizing problem along with a complete understanding of its long time behavior can be obtained by resorting to weak Young measure valued solutions in the spirit of Kinderlehrer and Pedregal (1992) [1] and Demoulini (1996) [2]. Solutions are unique (to an extent to be specified) but can exhibit “micro-oscillations” (in the sense of minimizing sequences and in the spirit of material science) between “preferred” gradient states. In the limit of vanishing regularization, the preferred gradients have size 0 or ∞ thus explaining the well-known phenomenon of staircasing. The theoretical results do completely confirm and/or predict numerical observations concerning the generic behavior of solutions.     

An analysis of the Perona‐Malik scheme

We investigate how the Perona‐Malik scheme evolves piecewise smooth initial data in one dimension. By scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size, we obtain a meaningful continuum limit. The resulting evolution can be seen as the gradient flow for an energy, just as the discrete evolutions are gradient flows for discrete energies. It involves, except at special isolated times, solving a system of heat equations coupled to each other through nonlinear boundary conditions. At the special times, the solutions experience gradient blowup; nevertheless, there is a natural continuation for the solutions beyond these singular times. © 2001 John Wiley & Sons, Inc.     

A Nonlinear Diffusion Equation-Based Model for Ultrasound Speckle Noise Removal

Ultrasound images are contaminated by speckle noise, which brings difficulties in further image analysis and clinical diagnosis. In this paper, we address this problem in the view of nonlinear diffusion equation theories. We develop a nonlinear diffusion equation-based model by taking into account not only the gradient information of the image, but also the information of the gray levels of the image. By utilizing the region indicator as the variable exponent, we can adaptively control the diffusion type which alternates between the Perona–Malik diffusion and the Charbonnier diffusion according to the image gray levels. Furthermore, we analyze the proposed model with respect to the theoretical and numerical properties. Experiments show that the proposed method achieves much better speckle suppression and edge preservation when compared with the traditional despeckling methods, especially in the low gray level and low-contrast regions.     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing

Summary. We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified (in the sense of Catté, Lions, Morel and Coll) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing). The proof is based on a-priori estimates and Kolmogorov's compactness theorem. The implementation aspects and computational results are discussed. Received January 7, 1999 / Revised version received May 31, 2000 / Published online March 20, 2001     

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.     

Image segmentation by using the localized subspace iteration algorithm

An image segmentation algorithm called"segmentation based on the localized subspace iterations"(SLSI)is proposed in this paper.The basic idea is to combine the strategies in Ncut algorithm by Shi and Malik in 2000 and the LSI by E,Li and Lu in 2007.The LSI is applied to solve an eigenvalue problem associated with the affinity matrix of an image,which makes the overall algorithm linearly scaled.The choices of the partition number,the supports and weight functions in SLSI are discussed.Numerical experiments for real images show the applicability of the algorithm.     

Neighborhood filters and PDE’s

Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona–Malik equation, one of the first nonlinear PDE’s proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.This work has been partially financed by the Centre National d’Etudes Spatiales (CNES), the Office of Naval Research under grant N00014-97-1-0839, the Ministerio de Ciencia y Tecnologia under grant MTM2005-08567. During this work, the first author had a fellowship of the Govern de les Illes Balears for the realization of his PhD.     

ON AVOIDANCE ACTIVITIES IN FISHERY ENFORCEMENT MODELS

Compliance and enforcement in fisheries are important issues from an economic point of view since management measures are useless without a certain level of enforcement. These conclusions come from the well‐established theoretical literature on compliance and enforcement problems within fisheries and a common result is that, it is efficient to set fines as high as possible and monitoring as low as possible, when fines are costless and offenders are risk neutral. However, this result is sensitive to the assumption that fishermen cannot engage in avoidance activities, e.g., activities to reduce the likelihood of being detected when noncomplying. The paper presents a model of fisheries that allows the fishermen to engage in avoidance activities. The conclusions from the model are that, under certain circumstances, fines are costly transfers to society since they not only have a direct positive effect on the level of deterrence, but also an indirect negative effect in the form of increased avoidance activities to reduce the probability of detection. The paper contributes to the literature on avoidance activities by introducing the externality from the illegal behavior as an endogenous effect on other offenders. For an externality, that has an exogenous effect on other actors, Malik shows that fines are only costly transfers for conditional deterrence (when one actor is deterred while another actor is not). For fisheries, we show that fines are also costly transfers under no deterrence (when no agents are deterred).     

Gradient estimates for the Perona–Malik equation

We consider the Cauchy problem for the Perona–Malik equation

A Poincaré–Birkhoff–Witt criterion for Koszul operads

The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to prove that an operad is Koszul. We define the notion of Poincaré–Birkhoff–Witt basis in the context of operads. Then we show that an operad having a Poincaré–Birkhoff–Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincaré–Birkhoff–Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie, Poisson) have a Poincaré–Birkhoff–Witt basis. We also prove by our methods that new operads are Koszul.     

Poincaré–Birkhoff–Witt Deformation of Koszul Calabi–Yau Algebras

The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained by He et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.     

On the generalized Ball bases

The Ball basis was introduced for cubic polynomials by Ball, and two different generalizations for higher degree m polynomials have been called the Said–Ball and the Wang–Ball basis, respectively. In this paper, we analyze some shape preserving and stability properties of these bases. We prove that the Wang–Ball basis is strictly monotonicity preserving for all m. However, it is not geometrically convexity preserving and is not totally positive for m>3, in contrast with the Said–Ball basis. We prove that the Said–Ball basis is better conditioned than the Wang–Ball basis and we include a stable conversion between both generalized Ball bases. The Wang–Ball basis has an evaluation algorithm with linear complexity. We perform an error analysis of the evaluation algorithms of both bases and compare them with other algorithms for polynomial evaluation.     

Deformations of Koszul Artin–Schelter Gorenstein algebras

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi–Yau algebra to be Calabi–Yau. The relations between the Calabi–Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul AS–Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi–Yau algebra is still Calabi–Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).