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In this paper, we propose a new three-stage segmentation framework based on a convex variant of the Mumford--Shah model and the intensity inhomogeneity information of an image. The first stage in o...

We reformulate the original phase retrieval problem into two variational models (with and without regularization), both containing a globally Lipschitz differentiable term. These two models can be efficiently solved via the proposed Partially Preconditioned Proximal Alternating Linearized Minimization (P${}^3$ALM) for masked Fourier measurements. Thanks to the Lipschitz differentiable term, we prove the global convergence of P${}^3$ALM for solving the nonconvex phase retrieval problems. Extensive experiments are conducted to show the effectiveness of the proposed methods.

In this paper, we focus on variational approaches for image inpainting in transform domain and propose two new algorithms, iterative coupled transform domain inpainting (ICTDI) and iterative decoupled transform domain inpainting. In the derivation of ICTDI, we use operator splitting and the quadratic penalty technique to get a new approximate problem of the basic model. By the alternating minimization method, the approximate problem can be decomposed as three relatively simple subproblems with closed-form solutions. However, ICTDI is not efficient when some adaptive regularization operator is used, such as the learned BM3D frame. To overcome this drawback, with some modifications, we decouple our framework into three relatively independent parts: denoising, linear combination in the transform domain, and linear combination in the image domain. Therefore, we can use any existing denoising method in the denoising step. We consider three choices for regularization operators in our approach: gradient operator...

The phase retrieval problem has drawn considerable attention, as many optical detection devices can only measure magnitudes of the Fourier transform of the underlying object (signal or image). This paper addresses the phase retrieval problem from incomplete data, where only partial magnitudes of Fourier transform are obtained. In particular, we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer-valued ones. Furthermore, we demonstrate theoretically and numerically that three diffracted sets of (complete) magnitude data are sufficient to recover the object. To compensate for incomplete information, we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties. The proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence. Numerical experiments valid...

We propose a weighted difference of anisotropic and isotropic total variation (TV) as a regularization for image processing tasks, based on the well-known TV model and natural image statistics. Due to the form of our model, it is natural to compute via a difference of convex algorithm (DCA). We draw its connection to the Bregman iteration for convex problems and prove that the iteration generated from our algorithm converges to a stationary point with the objective function values decreasing monotonically. A stopping strategy based on the stable oscillatory pattern of the iteration error from the ground truth is introduced. In numerical experiments on image denoising, image deblurring, and magnetic resonance imaging (MRI) reconstruction, our method improves on the classical TV model consistently and is on par with representative state-of-the-art methods.

In this paper, we propose a new fast variational approach to dehaze and denoise simultaneously. The proposed method first estimates a transmission map using a windows adaptive method based on the celebrated dark channel prior. This transmission map can significantly reduce the edge artifact in the resulting image and enhance the estimation precision. The transmission map is then converted to a depth map, with which the new variational model can be built to seek the final haze- and noise-free image. The existence and uniqueness of a minimizer of the proposed variational model is further discussed. A numerical procedure based on the Chambolle--Pock algorithm is given, and the convergence of the algorithm is ensured. Extensive experimental results on real scenes demonstrate that our method can restore vivid and contrastive haze- and noise-free images effectively.

In this paper, a two-stage method for segmenting blurry images in the presence of Poisson or multiplicative Gamma noise is proposed. The method is inspired by a previous work on two-stage segmentation and the usage of an I-divergence term to handle the noise. The first stage of our method is to find a smooth solution $u$ to a convex variant of the Mumford--Shah model where the $\ell_2$ data-fidelity term is replaced by an I-divergence term. A primal-dual algorithm is adopted to efficiently solve the minimization problem. We prove the convergence of the algorithm and the uniqueness of the solution $u$. Once $u$ is obtained, in the second stage, the segmentation is done by thresholding $u$ into different phases. The thresholds can be given by the users or can be obtained automatically by using any clustering method. In our method, we can obtain any $K$-phase segmentation ($K\geq 2$) by choosing $(K-1)$ thresholds after $u$ is found. Changing $K$ or the thresholds does not require $u$ to be recomputed. Exper...

In this paper, we study the image recovery problem where the observed image is simultaneously corrupted by blur and impulse noise. Our proposed patch-based model contains three terms: the sparse representation prior, the total variation regularization, and the data-fidelity term. We are interested in the two-phase approach. The first phase is to identify the possible impulse noise positions; the second phase is to recover the image via the patch-based model using noise position information. An alternating minimization method is then applied to solve the model. This approach works extremely well for image deblurring under salt-and-pepper noise. However, as the detection for random-valued noise is usually unreliable, extra work is then needed. Indeed, to get better recovery results for the latter case, we combine the two separate phases to simultaneously detect the random-valued noise positions and to recover the image. The numerical experiments clearly demonstrate the super performance of the proposed methods.

In this paper, we study the total variation structured total least squares method for image restoration. In the image restoration problem, the point spread function is corrupted by errors. In the model, we study the objective function by minimizing two variables: the restored image and the estimated error of the point spread function. The proposed objective function consists of the data-fitting term containing these two variables, the magnitude of error and the total variation regularization of the restored image. By making use of the structure of the objective function, an efficient alternating minimization scheme is developed to solve the proposed model. Numerical examples are also presented to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme.

The restoration of images corrupted by blurring and multiplicative noise is a challenging problem in applied mathematics that has attracted much attention in recent years. In this article, we propose a two-step approach to solve the problem of restoring images degraded by multiplicative noise and blurring, where the multiplicative noise is first reduced by nonlocal filters and then a convex variational model is adopted to obtain the final restored images. The variational model of the second step is composed of an $L_1$-$L_2$ data-fidelity term and a total variation regularization term. The alternating direction method (ADM) is utilized to solve this variational problem, and we also prove that the ADM algorithm converges at least linearly. Experimental results show that the proposed two-step approach performs better than the existing methods for restoring images with multiplicative noise and blurring, both in the quality of the restored images and the convergence speed of the algorithms.

The Mumford--Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two-stage segmentation method based on the Mumford--Shah model. The first stage of our method is to find a smooth solution $g$ to a convex variant of the Mumford--Shah model. Once $g$ is obtained, then in the second stage the segmentation is done by thresholding $g$ into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, $g$ can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle--Pock method. We prove that our method is convergent and that the solution $g$ is always unique. In our method, there is no need to specify the number of segments $K$ ($K\geq2$) before finding $g$. We can obtain any $K$-phase segmentations by choosing $(K-1)$ thresholds after $g$ is found in the first stage, and in the second...

The goal of this paper is to develop a multiphase image segmentation method based on fuzzy region competition. A new variational functional with constraints is proposed by introducing fuzzy membership functions which represent several different regions in an image. The existence of a minimizer of this functional is established. We propose three methods for handling the constraints of membership functions in the minimization. We also add auxiliary variables to approximate the membership functions in the functional such that Chambolle's fast dual projection method can be used. An alternate minimization method can be employed to find the solution, in which the region parameters and the membership functions have closed form solutions. Numerical examples using grayscale and color images are given to demonstrate the effectiveness of the proposed methods.