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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.

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.

Total variation (TV) regularization model has the excellent performance in noise-removing and edge-preserving. However, it often yields staircase artifacts in the smooth region of the restorations. To attack this problem, we propose a generalized hybrid nonconvex variational regularization model in this paper, which utilizes two nonconvex regularizers to impose the priors to the two different components of the images, respectively. One is nonconvex TV regularizer that measures the piecewise constant component of the input image, and the other is nonconvex Laplacian regularizer that models the piecewise smooth component. New model inherits the advantages of the nonconvex regularization and the first- and second-order hybrid variational regularization, which can well remove the noises while preserving edges and reducing staircase artifacts. To solve this nonconvex minimization problem efficiently, we propose a first-order algorithm based on alternating direction method of multipliers (ADMM) combining with majorization–minimization (MM) scheme. In addition, a sufficient condition for the convergence of the proposed algorithm is provided. In the experiments, we compare the proposed model with several state-of-the-art image denoising models, numerical results illustrate the effectiveness of the proposed model and algorithm for both synthetic and real images in terms of peak signal to noise ratio (PSNR) and structural similarity (SSIM) indexes.

In this paper, we propose a variational level set model for multiscale image segmentation that can extract desired object from the background under various scales. The new model consists of three terms: A boundary extraction term, a regularization term and a multiscale representation term. The boundary extraction term is based on the Chan–Vese (CV) functional, which drives the active contour toward the boundaries of the objects. The regularization term penalizes the length of the active contour and keeps the level set function close to a signed distance function , which enables the active contour to maintain its stability during the evolution process. The multiscale representation term is based on the total variation (TV) energy that can provide a multiscale representation of the input image by tuning the scale parameter. We use an alternating iterative algorithm that combines the gradient descent algorithm and the alternating direction method of multipliers (ADMM) to numerically solve the model. The experimental results for both synthetic and real images demonstrate the robustness and effectiveness of the proposed model for multiscale image segmentation. In addition, the proposed model is a promising model for image denoising applications.

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.

In this paper, we propose a variational model in binary level set framework for segmenting two-phase images with impulse noise. In this model, the contour is implicitly represented by the discontinuity of a binary level set function instead of the zero-level set. The energy functional consists of three terms, i.e., the data term that is defined by L1-norm metric, the regularization term that is defined by Dirichlet energy of level set function, and the penalty term that punishes the deviation of level set function from binary function. We design a three-step time-splitting scheme to solve the gradient descent flow of the proposed model efficiently, in which the flow equation is divided into three differential equations that are solved sequentially and alternatively until convergence. We present some experimental results that are performed on simulated and real images, which demonstrate that our model is very robust to impulse noise and has the best performance compared with five relevant models.

Haze removal (or dehazing) is very important for many applications in computer vision. Because depth information and atmospheric light are usually unknown in practice, haze removal is a challenging problem, especially for single image dehazing. In this paper, we propose a new variational model for removing haze from a single input image. The proposed model combines Koschmieder’s law with Retinex assumption that an image is the product of illumination and reflection. We assume that scene depth and surface radiance are spatially piecewise smooth, total variation is thus used for regularization in our model. The proposed model is defined as a constrained optimization problem, which is solved by an alternating minimization scheme and a fast gradient projection algorithm. Theoretical analyses are given for the proposed model and algorithm. Some numerical examples are presented, which have shown that our model has the best visual effect and the highest average PSNR (Peak Signal-to-Noise Ratio) compared to six relevant models in the literature.

In this paper, we present a new variant of level set methods and then propose a ternary variational level set model involving L0 gradient regularizer and L0 function regularizer in discrete framework, following the Chan-Vese model for image segmentation. Different from the existing level set methods, we use the 0.5-level set of a ternary function whose values are within {0, 0.5, 1} to implicitly represent the interfaces between subregions and use L0 counting operator to discretely measure the length of interfaces and the area of foreground subregions. The proposed model can be regarded as a discrete form of the Chan-Vese model. Based on the half-quadratic splitting method, we design an alternating minimization algorithm to solve our model efficiently. Experimental results show that the proposed method has good performance for segmentation of images with severe noise, outliers or low contrast.

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.

Display Omitted We propose a multiscale variational decomposition model.We apply the proposed model for hierarchical image restoration and reconstruction.We prove the nontrivial property and the convergence of our model.Experimental results verify the correctness of the theoretical analysis. Variational decomposition has been widely used in image denoising, however, it can't distinguish texture from noise well. Replacing the fixed parameter in the (BV, G) decomposition with a monotone increasing sequence, and iteratively taking the residual of the previous step as the input to decompose, we propose a multiscale variational decomposition model in this paper. Unlike the fixed-scale decomposition, the new model can decompose the input image into a sum of a series of features with different scales. So, texture can be distinguished from noise. In addition, we prove the nontrivial property and the convergence of this multiscale decomposition, and introduce a hybrid iteration algorithm that combines the first-order primal-dual algorithm with the gradient decent method to numerically solve the multiscale decomposition model. Numerical results validate the effectiveness of the proposed model. Furthermore, we apply this multiscale decomposition for image hierarchical restoration. Compared with the classical hierarchical (BV, L2) decomposition, hierarchical wavelet decomposition and fixed-scale (BV, G) decomposition, our model has better performance for both synthetic and real images in terms of PSNR and MSSIM.

We propose a hierarchical (BV, G) variational decomposition model for multiscale texture extraction in this paper, which can offers a hierarchical, separated representation of image texture in different scales. The proposed hierarchical decomposition is obtained by replacing the fixed scale parameter of the A2BC model with a varying sequence. Some properties of this hierarchical decomposition are presented and its convergence is proved. We adopt Euclidean projection algorithm to solve this hierarchical decomposition model numerically. In addition, we use this hierarchical decomposition to achieve the multiscale texture extraction. The performance of the proposed model is demonstrated with both synthetic and real images.

In this paper, we present a binarization method for degraded document images with contrast variation or low contrast, which couples a selective shock filter and a time-dependent source to a fourth order indirect diffusion process. The role of the selective shock filter is to enhance the edges as far as possible, while the time-dependent source plays an increasingly predominant role in the process of binarization. The fourth order indirect diffusion not only keeps the merit of noise immunity of the direct diffusion counterpart, but also preferably preserves the low-frequency components to avoid blur and dislocation of weak edges. We develop a novel numerical scheme for solving our model effectively and efficiently, by incorporating finite differencing with the parallel splitting up method in the literature. The proposed algorithm is tested on seven publicly available datasets (Document Image Binarization COmpetition (DIBCO) 2009 to 2014 and 2016). The experimental results indicate that our model is very effective for binarization of document images with contrast variation or low contrast, and has mostly the best effect for document images from the DIBCO datasets, compared with four partial differential equation (PDE) models and four benchmark and recent non-PDE 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 present a fractional order total variation (TV) regularization functional for image super-resolution, the role of which is to better handle the texture details of image. This regularization functional is then incorporated into a variational formulation with an image fidelity term and the usual TV regularization that can efficiently preserve the discontinuities and image structures. The resulting evolution equation is the gradient descent flow that minimizes the overall functional. The proposed model has been applied to eight real images with promising results; unlike the existing TV-based image super-resolution models, the proposed model does not suffer from block artifacts, staircase edges and false edge near the edges.

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.

We propose a new active contour model which integrates a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model not only can provide desirable segmentation results in the presence of intensity inhomogeneity but also allows for more flexible initialization of the contour compared to the RSF and LIF models, and we give a theoretical proof to compute a unique steady state regardless of the initialization; that is, the convergence of the zero-level line is irrespective of the initial function. This means that we can obtain the same zero-level line in the steady state, if we choose the initial function as a bounded function. In particular, our proposed model has the capability of detecting multiple objects or objects with interior holes or blurred edges.

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 present a scheme of improvement on the region-scalable fitting (RSF) model proposed by Li et al. (Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing 17(10) (2008) 1940-1949) in terms of robustness to initialization and noise. First, the Gaussian kernel for the RSF energy is replaced with a ''mollifying'' kernel with compact support. Second, the RSF energy is redefined as a weighted energy integral, where the weight is local entropy deriving from a grey level distribution of image. The total energy functional is then incorporated into a variational level set formulation with two extra internal energy terms. The new RSF model not only handles better intensity inhomogeneity, but also allows for more flexible initialization and more robustness to noise compared to the original RSF model.

We propose anLp(|∇I|)-based adaptively active contours model for image segmentation which is derived from the well-known Chan-Vese (C-V) model. Unlike the C-V model, the proposed model uses theLp(|∇I|) (p(|∇I|)>2)norm instead of theL2norm to define the external energy and incorporates an extra internal energy into the overall energy. Due to the variable exponentp(|∇I|) which could fit the image gradient information adaptively, the proposedLp(|∇I|)-based model has the hope of segmenting those images with low contrast and blurred boundaries. Experimental results show that the proposed model withp(|∇I|)>2really can effectively and quickly segment those images with low contrast and blurred boundaries.

A tensor diffusion level set method is presented to extract infrared (IR) targets contour under a sky–mountain–water complex background. The proposed model combines tensor diffusion operator and the eigenvalues of tensor-image into a common energy minimization level set framework. By incorporating the information of image tensor diffusion operator into the external energy term, the level set function can move in a specific way. And eigenvalues of tensor-image are used for the regularization of zero level curves in order to diminish the influence of image ‘clutter’ and noise. An additional benefit of the proposed method is robust to initial conditions. Experimental results show very good performance of the tensor diffusion level set method for IR targets contours extraction.

An adaptive regularized level set method for image segmentation is proposed. A weightedp(x)-Dirichlet integral is presented as a geometric regularization on zero level curve, which is used to diminish the influence of image noise on level set evolution while ensuring the active contours not to pass through weak object boundaries. The idea behind the new energy integral is that the amount of regularization on the zero level curve can be adjusted automatically by the variable exponentp(x)to fit the image data. This energy is then incorporated into a level set formulation with an external energy term that drives the motion of the zero level set toward the desired objects boundaries, and a level set function regularization term that is necessary for maintaining stable level set evolution. The proposed model has been applied to a wide range of both real and synthetic images with promising results.

Following the recently proposed total variation (TV) up-sampling method, we present a novel image up-sampling algorithm based on fractional-order bidirectional diffusion. For the bidirectional diffusion, the forward diffusion occurs on the light side of edge, while the backward diffusion proceeds on the dark side. This bidirectional diffusion can reduce the edge width and avoid the appearances of false edge or texture and block effects. Moreover, the fractional-order derivative is used to avoid strong contrast near the edge in the interpolated images. The experiments show that, unlike the TV up-sampling (TVUP) method, the proposed algorithm does not suffer from the drift of edges, block effect in the smooth regions, false edges, and false texture.

We propose a novel variational formulation (external energy) for implicit active contours that forces the level-set function to have the opposite sign along the edges at convergence. This external energy is then incorporated into a variational level-set formulation with two extra regularization terms (internal energy). The resulting evolution of the level-set function is the gradient flow that minimizes the overall energy functional. Because of the external energy, the level-set function can be initialized to any bounded function (e.g., a constant function), which completely eliminates the need of initial contours. This implies that the new formulation is robust to initialization or even free of manual initialization. The proposed model is been applied to both real and synthetic images with promising results. Besides, our model is especially appropriate for segmentation of real images with a complex and /or nonuniform background.

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.

We propose a novel image segmentation algorithm using piecewise smooth (PS) approximation to image. The proposed algorithm is inspired by four well-known active contour models, i.e., Chan and Vese’ piecewise constant (PC)/smooth models, the region-scalable fitting model, and the local image fitting model. The four models share the same algorithm structure to find a PC/smooth approximation to the original image; the main difference is how to define the energy functional to be minimized and the PC/smooth function. In this article, pursuing the same idea we introduce different energy functional and PS function to search for the optimal PS approximation of the original image. The initial function with our model can be chosen as a constant function, which implies that the proposed algorithm is robust to initialization or even free of manual initialization. Experiments show that the proposed algorithm is very appropriate for a wider range of images, including images with intensity inhomogeneity and infrared ship images with low contrast and complex background.