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Binarization has always been a very challenging task for degraded document images because of the variety and complexity of degradations. This paper proposed a nonlinear diffusion equation with selective source for restoration of degraded document images, followed by a binary projection for binarization. The source is composed of two parts: one is used for the restoration of contaminated background, and another is responsible for the fidelity of texts. The evolution equation is numerically solved by the simplest finite differencing. The proposed method is tested on publicly available datasets (DIBCO (Document Image Binarization Competition) 2009–2014, 2016) and is compared with five models based on partial differential equation (PDE). The experimental results show that the proposed method is effective for degraded document images and has achieved generally the best performance of binarization compared to the other five PDE methods.

In this paper, we propose a noise-robust online convolutional coding model for image representation, which can use the noisy images as training data. Then an alternating algorithm is utilized to convert the model into two sub-problems, the image pursuit problem and the dictionary learning problem. For the image pursuit problem, the Gauss elimination method is used to solve the equation set which is derived by the Euler equation and discrete Fourier transform. For the dictionary learning problem, a gradient-descent flow is derived to solve it. Experimental results show that our method can output more meaningful feature representations compared to the related models while the training data was corrupted by Poisson noise.

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.

This paper presents an adaptive enhancement model with selective smoothing for restoration of degraded document images with blur, noise and bleed-through, which involves adaptive shock filtering and selective diffusion processes. A novel hybrid scheme is developed to solve the proposed model numerically, which combines explicit finite difference and exponential smoothing. Numerical experiments show that the proposed model is very effective for restoration of degraded document images with blur, noise and bleed-through, and has averagely the best performance on the DIBCO (Document Image Binarization Competition) series datasets, compared to five PDE (partial differential equation)-based models for restoration of degraded document images.

In this paper, a vector total fractional-order variation (VTV-β) is proposed. Then, VTV-β model and Gβ(Ω) model are proposed for color image denoising and decomposition, respectively. Some properties of the VTV-β are investigated and an alternative algorithm is used to solve the two models. Some experimental results are given to show the effectiveness and advantages of our methods.

This paper proposes a nonlinear edge-preserving diffusion equation with an adaptive source term for binarization of degraded document images. The role of nonlinear diffusion term is to smooth images with preservation of text edges and corners, while the source term is responsible for the desired binarization. Unlike other binarization techniques (such as clustering-based and threshold-based), the idea behind the proposed method is that a sequence of gradually binarized images is obtained by solving the evolution equation starting with the image to be binarized, and tends to the slightly smoothed version of the desired binary image at infinity. A semi-implicit parallel splitting-up method is developed for solving the proposed model effectively. The proposed model with algorithm is tested on the DIBCO (Document Image Binarization Competitions) series datasets. The results show that it has generally the best performance, compared to four PDE (partial differential equation)-based binarization models, and six recent and benchmark binarization algorithms (non-PDE based).

In this paper, we put forward an idea of indirect diffusion and further develope an indirect diffusion-based level set model for image segmentation. This model is based on the dynamic process of diffusion that is posed indirectly on level set function by way of auxiliary function, coupled with a transition region-based force that exhibits the desired sign-changing property. It is formulated as a coupled system of two evolution equations, in which the first equation drives the motion of zero level set toward the object edges and makes it possible to set a termination criterion on the algorithm, while the second equation (indirect diffusion) smoothens the auxiliary function and keeps the auxiliary function as close to the level set function as possible. The derived model can effectively be solved purely by the simplest explicit finite difference. Experimental results show that the proposed model not only has the strong capability of noise immunity, but it also can much better conduce to extraction of deeply concave edges and preservation of sharp corners, compared with the direct diffusion-based counterpart.

The segmentation of images with severe noise has always been a very challenging task because noise has great influence on the accuracy of segmentation. This paper proposes a robust variational level set model for image segmentation, involving the kernel metric based on the Gaussian radial basis function (GRBF) kernel as the data fidelity metric. The kernel metric can adaptively emphasize the contribution of pixels close to the mean intensity value inside (or outside) the evolving curve and so reduce the influence of noise. We prove that the proposed energy functional is strictly convex and has a unique global minimizer in B V ( Ω ) . A three-step time-splitting scheme, in which the evolution equation is decomposed into two linear differential equations and a nonlinear differential equation, is developed to numerically solve the proposed model efficiently. Experimental results show that the proposed method is very robust to some types of noise (namely, salt & pepper noise, Gaussian noise and mixed noise) and has better performance than six state-of-the-art related models.

In this letter, we present a fast and effective algorithm for solving the Poisson-modified total variation model proposed in [Le et al. , “A variational approach to reconstructing images corrupted by Poisson noise,” J. Math. Imag. Vis. , vol. 27, no 3, pp. 257–263, Apr. 2007]. The existence and uniqueness of solution for the model are proved by using a different method. A semi-implicit difference scheme is designed to discretize the derived gradient descent flow with a large time step. Different from the original numerical scheme, our scheme is conditional stable with a less stringent condition and can ensure that the numerical solution is strictly positive in image domain. Experimental results show the efficiency and effectiveness of our algorithm.

In this paper, we propose a variational level set model with indirect regularization term for image segmentation. Instead of using direct regularization on level set function, we introduce an auxiliary function to regularize indirectly the level set function. Our energy functional consists of a data term, a link term of level set function with the auxiliary function and a regularization term of the auxiliary function. We prove that the energy functional is convex in L 2 ( ? ) × W 1 , 2 ( ? ) and give the convergence analysis of the alternating minimization algorithm that we utilized. We show that the indirect regularization has some advantages over direct regularization theoretically and experimentally. Experimental results illustrate that the proposed model can better handle images with high noise, angle and weak edges. Present a novel indirectly regularized variational level set model.We give a rigorously analytical study on the proposed model.Present an alternating minimization algorithm and give its convergence analysis.The indirect regularization term can be easily integrated into existing variational level set methods.

We propose an image segmentation model that is derived from reaction---diffusion equations and level set methods. In our model, a diffusion term is used for regularization of a level set function, and a reaction term has the desired sign property to force the level set function to move up or down and finally identify an object and its background. Our level set function can be initialized to any bounded function (e.g., a constant function). The proposed model can be applied to a wider range of images with promising results, especially for real images that have high noise and blurred boundaries. This study gives a new method for the further investigations of reaction---diffusion equations directly for segmentation.

In this paper, we propose a strictly convex energy functional in a level set formulation for the purpose of two-phase image segmentation. We prove that the value of the unique global minimizer for the energy functional is within the interval -1, 1 for any image, and equals to 1 in the object and -1 in the background for an ideal binary image. A pointwise convergent numerical scheme is presented to solve the gradient descent flow equation. The proposed model is allowed for flexible initialization and can set a reasonable termination criterion on the algorithm. The proposed model has been successfully applied to some synthesized and real images with promising results. HighlightsThe proposed energy functional is strictly convex in L2-Function space.We give a rigorously analytical study on the proposed model.The proposed model is allowed for constant initialization.We present a numerical scheme and prove its pointwise convergence analytically.

This paper proposes a variational model with barriers for Retinex, borrowing the ideas of barrier methods. We first present an energy functional and then deduce a new energy functional from it by adding two barriers. The proposed model is defined as a constrained optimization problem associated with the deduced energy functional. Next, an alternating minimization scheme is used to solve the proposed model. Some theoretic analyses are given for the proposed model and algorithm. Finally, numerical examples are presented to show the effectiveness of the proposed model with its algorithm.

The purpose of this paper is to discuss inequalities related to operator versions of the classical Bohr inequality. We obtain refinements of some inequalities due to Cheung and Pecaric [J. Math. Anal. Appl. 323 (2006) 403–412] and Zhang [J. Math. Anal. Appl. 333 (2007) 1264–1271]. Moreover, we present two inequalities for multiple operators, which are similar to ones proposed by Chansangiam et al. [J. Math. Anal. Appl. 356 (2009) 525–536]. Mathematics subject classification (2010): 47A63.

Level set evolution without re-initialization is a novel variational level set method for edge contour extraction, which has many advantages over the traditional level set formulations. The resulting evolution equation is derived from the minimization of an energy functional which is a linear combination of a weighted length, weighted area and deviation penalization energy. However, the zero level set always keeps evolving in only one direction during evolution, depending on the sign of the scale parameter associated with the weighted area. This inconvenient makes the evolution highly sensitive to the contour initializations. In this paper, we propose an adaptive level set evolution equation following this method, wherein the scale parameter associated with the weighted area is modified as an adaptive variable sign function and one of two terms associated with the weighted length is removed to reduce computational cost. The proposed equation avoids completely the intrinsic limitation mentioned above and offers many advantages over the original equation, as illustrated by several examples of contour extraction, such as robustness to noise and detection of objects with discontinuous boundaries.

In order to achieve a mutiscale representation and texture extraction for textured image, a hierarchical(BV,Gp,L2)decomposition model is proposed in this paper. We firstly introduce the proposed model which is obtained by replacing the fixed scale parameter of the original(BV,Gp,L2)decomposition with a varying sequence. And then, the existence and convergence of the hierarchical decomposition are proved. Furthermore, we show the nontrivial property of this hierarchical decomposition. Finally, we introduce a simple numerical method for the hierarchical decomposition, which utilizes gradient decent for energy minimization and finite difference for the associated gradient flow equations. Numerical results show that the proposed hierarchical(BV,Gp,L2)decomposition is very appropriate for multiscale representation and texture extraction of textured image.

In this paper, we first present a hierarchical (BV,G p ,L 2) variational decomposition model and then use it to achieve multiscale texture extraction which offers a hierarchical, separated representation of image texture in different scales. The starting point is the use of the variational (BV,G p ,L 2) decomposition; a given image f?L 2(?) is decomposed into a sum of u 0+v 0+r 0, where (u 0,v 0)?(BV(?),G p (?)) is the minimizer of an energy functional E(f,? 0;u,v) and r 0 is the residual (i.e. r 0=f?u 0?v 0). In this decomposition, v 0 represents the fixed scale texture of f, which is measured by the parameter ? 0. To achieve a multiscale representation, we proceed to capture essential textures of f which have been absorbed by the residuals. Such a goal can be achieved by iterating a refinement decomposition to the residual of the previous step, i.e. r i =u i+1+v i+1+r i+1, where (u i+1,v i+1) is the minimizer of E(r i ,? 0/2 i+1;u,v). In this manner, we can obtain a hierarchical representation of f. In addition, we discuss some theoretical properties of the hierarchical (BV,G p ,L 2) decomposition and give its numerical implementation. Finally, we apply this hierarchical decomposition to the multiscale texture extraction. The performance of this method is demonstrated with both synthetic and real images.

Variational level set methods for image segmentation involve minimizing an energy functional over a space of level set functions using a continuous gradient descent method. The functional includes the internal energy (curve length, usually) for regularization and the external energy that aligns the curves with object boundaries. Current practice is, in general, to minimize the energy functional by calculating the L 2 gradient of the total energy. However, the Sobolev gradient is particularly effective for minimizing the curve length functional by the gradient descent method in that it produces the solution in a single iteration. In this paper, we thus propose to use the Sobolev gradient for the internal energy (curve length), while still using the L 2 gradient for the external energy. The test results show that the “ L 2 plus Sobolev” gradient scheme is significantly more computationally efficient than the methods only based on the L 2 gradient.

This paper aims to present some inequalities for unitarily invariant norms. We first give inverses of Young and Heinz type inequalities for scalars. Then we use these inequalities to establish some inequalities for unitarily invariant norms. Mathematics subject classification (2010): 15A45, 15A60.

In this note, we give some refinements of Fischer-type determinantal inequalities for accretive-dissipative matrices which are due to Lin (Linear Algebra Appl. 438:2808-2812, 2013) and Ikramov (J. Math. Sci. (N.Y.) 121:2458-2464, 2004).

In this paper, we establish some new inequalities of Simpson’s type based on -convexity for differentiable mappings. This contributes to new better estimates than presented already. Some applications for special means of real numbers and error estimates for some numerical quadrature rules are also given. MSC:26D15, 26D10.

The purpose of the paper is to present some inequalities for eigenvalues of positive semidefinite matrices.

In this paper, we propose a novel level set evolution model in a partial differential equation (PDE) formulation. According to the governing PDE, the evolution of level set function is controlled by two forces, an adaptive driving force and a total variation (TV)-based regularizing force that smoothes the level set function. Due to the adaptive driving force, the evolving level set function can adaptively move up or down in accordance with image information as the evolution proceeds forward in time. As a result, the level set function can be simply initialized to a constant function rather than the widely-used signed distance function or piecewise constant function in existing level set evolution models. Our model completely eliminates the needs of initial contours as well as re-initialization, and so avoids the problems resulted from contours initialization and re-initialization. In addition, the evolution PDE can be solved numerically via a simple explicit finite difference scheme with a significantly larger time step. The proposed model is fast enough for near real-time segmentation applications while still retaining enough accuracy; in general, only a few iterations are needed to obtain segmentation results accurately.

In this paper, we propose a variational segmentation model to deal with intensity inhomogeneity and Poisson noise. An energy functional is first proposed, which uses a data-fidelity term deduced from Poisson distribution instead of the usual L2 norm as a measure of fidelity. Due to the new data-fidelity measure, this energy functional can fit the image intensity more accurately while it can diminish the influence of Poisson noise on segmentation results. We then reformulate the energy function as globally convex formulation, which allows for more flexible initialization. The final convex energy functional is minimized via the dual formulation instead of the usually used gradient descent method. Experimental results show that the proposed model can efficiently segment images with intensity inhomogeneity and Poisson noise.