A Retinex-based total variation approach for image segmentation and bias correction

Image segmentation methods usually suffer from intensity inhomogeneity problem caused by many factors such as spatial variations in illumination (or bias fields of imaging devices). In order to address this problem, this paper proposes a Retinex-based variational model for image segmentation and bias correction. According to Retinex theory, the input inhomogeneous image can be decoupled into illumination bias and reflectance parts. The main contribution of this paper is to consider piecewise constant of the reflectance, and thereby introduce the total variation term in the proposed model for correcting and segmenting the input image. This is different from the existing model in which the spatial smoothness of the illumination bias is employed only. The existence of the minimizers to the variational model is established. Furthermore, we develop an efficient algorithm to solve the model numerically by using the alternating minimization method. Our experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.     

A detail preserving variational model for image Retinex

In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments.     

基于PICCS的非局部TGV能谱CT重建算法

能谱CT将宽谱划分为窄谱,导致通道内光子数目明显减少,加大了噪声影响,故从噪声投影中重建出高质量图像是能谱CT的一个研究热点.传统全变分(total variational,TV)容易造成重建图像中出现块状伪影等问题,总广义全变分(total generalized variation,TGV)算法可以逼近任意阶函数,再结合非局部均值算法的思想,同时考虑到不同能谱通道下重建图像的相关性,将高质量全能谱重建图像作为先验图像指导能谱CT重建,提出了基于先验图像约束压缩感知(prior image constrained compressed sensing,PICCS)的非局部TGV重建算法.实验结果表明,所提算法在抑制噪声的同时能够有效复原图像细节及边缘信息,且收敛速度快.     

Non-convex and non-smooth variational decomposition for image restoration

The variational image decomposition model decomposes an image into a structural and an oscillatory component by regularization technique and functional minimization. It is an important task in various image processing methods, such as image restoration, image segmentation, and object recognition. In this paper, we propose a non-convex and non-smooth variational decomposition model for image restoration that uses non-convex and non-smooth total variation (TV) to measure the structure component and the negative Sobolev space $H^{-1}$ to model the oscillatory component. The new model combines the advantages of non-convex regularization and weaker-norm texture modeling, and it can well remove the noises while preserving the valuable edges and contours of the image. The iteratively reweighted l¹ (IRL1) algorithm is employed to solve the proposed non-convex minimization problem. For each subproblem, we use the alternating direction method of multipliers (ADMM) algorithm to solve it. Numerical results validate the effectiveness of the proposed model for both synthetic and real images in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity index (MSSIM).     

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u ₀ into a sum u+v. Here u∈BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H ^−s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.     

Principal Component Analysis of Two-Dimensional Functional Data

This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online.     

Vector total fractional-order variation and its applications for color image denoising and decomposition

In this paper, a vector total fractional-order variation (VTV-β) is proposed. Then, VTV-β model and G^β(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.     

Minimizers for nonlocal perimeters of Minkowski type

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincaré–Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.     

A variational model with hybrid Hyper-Laplacian priors for Retinex

Retinex aims at estimating real reflectance images by removing the effect of illumination. We propose a nonconvex variational model for Retinex with novel priors for reflectance and illumination. Based on the statistics of the gradients of reflectance and illumination, we use the hyper-Laplacian prior to characterize the gradients of reflectance, and the hybrid hyper-Laplacian and Tikhonov prior to characterize the gradients of illumination. An efficient alternating direction method of multipliers (ADMM) is developed to solve the proposed model. Extensive numerical experiments show that the proposed method is comparable to the state-of-the-art methods quantitatively and qualitatively.     

Nonlocal patches based Gaussian mixture model for image inpainting

We consider the inpainting problem for noisy images. It is very challenge to suppress noise when image inpainting is processed. An image patches based nonlocal variational method is proposed to simultaneously inpainting and denoising in this paper. Our approach is developed on an assumption that the small image patches should be obeyed a distribution which can be described by a high dimension Gaussian Mixture Model. By a maximum a posteriori (MAP) estimation, we formulate a new regularization term according to the log-likelihood function of the mixture model. To optimize this regularization term efficiently, we adopt the idea of the Expectation Maximization (EM) algorithm. In which, the expectation step can give an adaptive weighting function which can be regarded as a nonlocal connections among pixels. Using this fact, we built a framework for non-local image inpainting under noise. Moreover, we mathematically prove the existence of minimizer for the proposed inpainting model. By using a splitting algorithm, the proposed model are able to realize image inpainting and denoising simultaneously. Numerical results show that the proposed method can produce impressive reconstructed results when the inpainting region is rather large.     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.     

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H^-1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349-370] which is constructed by the total variation and H^-1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler-Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268].     

Kinetic-fluid derivation and mathematical analysis of a nonlocal cross-diffusion–fluid system

In this article we propose a nonlocal cross-diffusion–fluid system describing the dynamics of multiple interacting populations living in a Newtonian fluid. First, we derive our nonlocal cross-diffusion–fluid system from a nonlocal kinetic-fluid model by the micro-macro decomposition method. Second, we prove the existence of weak solutions for the proposed system by applying the nonlinear Galerkin method with a priori estimates and compactness arguments. On the basis of micro-macro decomposition, we propose and develop an asymptotic-preserving numerical scheme. Finally, we discuss the computational results for the proposed system.     

Total image constrained diffusion tensor for spectral computed tomography reconstruction

Photon counting detector (PCD)-based spectral computed tomography (CT) is a promising imaging technique that enables high energy resolution imaging with narrow energy bins. However, the image quality is degraded because the number of photons in each energy bin is less than the number of photons in the full spectrum. To reconstruct high quality spectral CT images with narrow energy bins, we developed a total image constrained diffusion tensor (TICDT) for statistical iterative reconstruction (SIR) based on a penalized weighted least-squares (PWLS) principle, which is called “PWLS-TICDT.” Specifically, TICDT uses supplementary information from a high-quality total image as a structural prior for SIR, so that the narrow energy bin image can be enhanced, while some primary features are preserved. We also developed an alternating minimization algorithm to solve the associated objective function. We conducted qualitative and quantitative studies to validate and evaluate the PWLS-TICDT method using digital phantoms and preclinical data. Results from both numerical simulation and real PCD data studies show that the proposed PWLS-TICDT method achieves noticeable gains over competing methods in terms of suppressing noise, detecting low contrast objects, and preserving resolution. More importantly, the multi-energy images reconstructed by PWLS-TICDT method can generate more accurate basis material decomposition results than the other methods.     

Cahn–Hilliard equation with nonlocal singular free energies

Annali di Matematica Pura ed Applicata (1923 -) - We consider a Cahn–Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is...     

New normalized nonlocal hybrid level set method for image segmentation

This article introduces a new normalized nonlocal hybrid level set method for image segmentation. Due to intensity overlapping, blurred edges with complex backgrounds, simple intensity and texture information, such kind of image segmentation is still a challenging task. The proposed method uses both the region and boundary information to achieve accurate segmentation results. The region information can help to identify rough region of interest and prevent the boundary leakage problem. It makes use of normalized nonlocal comparisons between pairs of patches in each region, and a heuristic intensity model is proposed to suppress irrelevant strong edges and constrain the segmentation. The boundary information can help to detect the precise location of the target object, it makes use of the geodesic active contour model to obtain the target boundary. The corresponding variational segmentation problem is implemented by a level set formulation. We use an internal energy term for geometric active contours to penalize the deviation of the level set function from a signed distance function. At last, experimental results on synthetic images and real images are shown in the paper with promising results.     

A splitting primal-dual proximity algorithm for solving composite optimization problems

Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.     

A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising

The total variation model proposed by Rudin, Osher and Fatemi performs very well for removing noise while preserving edges. However, it favors a piecewise constant solution in BV space which often leads to the staircase effect, and small details such as textures are often filtered out with noise in the process of denoising. To preserve the textures and eliminate the staircase effect, we improve the total variation model in this paper. This is accomplished by the following steps: (1) we define a new space of functions of fractional-order bounded variation called the BV_α space by using the Grünwald–Letnikov definition of fractional-order derivative; (2) we model the structure of the image as a function belonging to the BV_α space, and the textures in different scales as functions belonging to different negative Sobolev spaces. Thus, we propose a class of fractional-order multi-scale variational models for image denoising. (3) We analyze some properties of the fraction-order total variation operator and its conjugate operator. By using these properties, we develop an alternation projection algorithm for the new model and propose an efficient condition of the convergence of the algorithm. The numerical results show that the fractional-order multi-scale variational model can improve the peak signal to noise ratio of image, preserve textures and eliminate the staircase effect efficiently in the process of denoising.     

Lorentz-Covariant Ultradistributions, Hyperfunctions, and Analytic Functionals

We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin–statistics theorem to nonlocal field theory.     

Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.