Multiple Degree Total Variation (MDTV) Regularization for Image Restoration.

We extend the novel higher degree total variation (HDTV) image regularization penalties to 3-D signals. These penalties generalize the popular total variation (TV) penalty to incorporate higher degree image derivatives. We adapt a fast alternating minimization algorithm designed for solving 2-D image recovery problems with HDTV regularization to the 3D setting. Numerical experiments on the compressed sensing recovery of 3-D magnetic resonance images show that HDTV and generalized HDTV improve the image quality significantly compared with TV. We also investigate the relationship between the recently introduced Hessian Schatten-norms and HDTV.

We introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation (TV) scheme. Motivated by novel reinterpretations of the classical TV regularizer, we derive two families of functionals involving higher degree partial image derivatives; we term these families as isotropic and anisotropic higher degree TV (HDTV) penalties, respectively. The isotropic penalty is the L(1) - L(2) mixed norm of the directional image derivatives, while the anisotropic penalty is the separable L(1) norm of directional derivatives. These functionals inherit the desirable properties of standard TV schemes such as invariance to rotations and translations, preservation of discontinuities, and convexity. The use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel, thus encouraging isotropic smoothing. In contrast, the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities, while continuing to smooth along the linelike features; this scheme thus enhances the linelike image characteristics analogous to standard TV. We also introduce efficient majorize-minimize algorithms to solve the resulting optimization problems. The numerical comparison of the proposed scheme with classical TV penalty, current second-degree methods, and wavelet algorithms clearly demonstrate the performance improvement. Specifically, the proposed algorithms minimize the staircase and ringing artifacts that are common with TV and wavelet schemes, while better preserving the singularities. We also observe that anisotropic HDTV penalty provides consistently improved reconstructions compared with the isotropic HDTV penalty.

Regularization plays a crucial role in reliably utilizing imaging systems for scientific and medical investigations. It helps to stabilize the process of computationally undoing any degradation caused by physical limitations of the imaging process. In the past decades, total variation regularization played a dominant role in the literature. Two forms of total variation regularizations, namely the first-order and the second-order total variation (TV-1 and TV-2) have been widely used. TV-1 has a disadvantage: it reconstructs images in the form of piece-wise constants when the noise and/or under-sampling is severe, while TV-2 reconstructs natural-looking images under such scenarios. On the other hand, TV-1 can recover sharp jumps better than TV-2. Two forms of generalizations, namely Hessian-Schatten norm (HSN) regularization, and total generalized variation (TGV) regularization, have been proposed and have become significant developments in the area of regularization for imaging inverse problems owing to their performance. While the strength of TGV is that it can combine the advantages of TV-1 and TV-2, HSN has better structure-preserving property. Here, we develop a novel regularization for image recovery that combines the strengths of TGV and HSN. We achieve this by restricting the maximization space in the dual form of HSN in the same way that TGV is obtained from TV-2. We call the new regularization the generalized Hessian-Schatten norm regularization (GHSN). We develop a novel computational method for image reconstruction using the new form of regularization based on the well-known framework called the alternating direction method of multipliers (ADMM). We demonstrate the strength of the GHSN using some reconstruction examples.

This paper, we propose a method that exploit total generalized variation (TGV) and Shearlet transform for image restoring. The TGV adaptive regularize different image regions at different levels and the Shearlet transform can efficiently represent image anisotropic features such as edges, curves. A new image restoration model combining TGV and Shearlet transform is proposed for image restoration. The proposed model is solved by splitting variables and applying the alternating direction method of multiplier (ADMM). Experimental results show that the proposed algorithm can effectively restore image and improve the quality.

Isotropic and anisotropic filtering norm-minimization: A generalization of the TV and TGV minimizations using NESTA

The preservation of discontinuous edges in seismic dataset such as faults, fractures, pinch-outs, channels and salt bodies while attenuating random noise is of great significance for the following seismic data interpretation. Noise reduction using inversion with total variation (TV) regularization is a kind of 2D edge-preserving filtering method. However, TV regularization is only limited to 2D data. In this paper, we extend the 2D method to 3D case by firstly taking difference of seismic data along different directions, and then performing Cauchy regularization. A synthetic 3D dataset, a real stacked OBC dataset and a real stacked land dataset are used to illustrate the performance of the method. The results show that the 3D seismic noise-reduction method we proposed can perfectly preserve edges of 3D discontinuous events while reducing random noise. In addition, there is no obvious trail of noise along the spatial directions since 3D noise-reduction scheme is adopted.

Total variation was recently introduced in many different magnetic resonance imaging applications. The assumption of total variation is that images consist of areas, which are piecewise constant. However, in many practical magnetic resonance imaging situations, this assumption is not valid due to the inhomogeneities of the exciting B1 field and the receive coils. This work introduces the new concept of total generalized variation for magnetic resonance imaging, a new mathematical framework, which is a generalization of the total variation theory and which eliminates these restrictions. Two important applications are considered in this article, image denoising and image reconstruction from undersampled radial data sets with multiple coils. Apart from simulations, experimental results from in vivo measurements are presented where total generalized variation yielded improved image quality over conventional total variation in all cases.

The weighted total generalized variation (TGV) is deflned and the Mumford-Shah model based on weighted TGV is proposed, in which the second-order weighted TGV semi-norm of images is used as the regularization term. Besides, the second-order weighted TGV semi-norm of the level set function is used for approximating the length of boundary. A numerical calculation model is presented for solving the unknown functions by using the alternating Split-Bregman method, Fenchel dual method, and FISTA (fast iterative shrinkage-thresholding algorithm), separately. Simulation results show that the second-order weighted TGV semi-norm of images has better denoising efiect than the common L2 norm of gradient norm and the weighted TV semi-norm. And the result of edge detection is better than the traditional TV semi-norm and weighted TV semi-norm by using the second-order weighted TGV semi-norm of the level set function to approximate the length of boundary.

The quantitative analysis of the blood flow in the brain tissue is one of the important problems in neurosurgery. It arises when diagnosing acute ischemic stroke. This problem can be solved using computed tomography (CT) perfusion imaging. There are various methods for extracting quantitative characteristics of cerebral blood flow from CT perfusion data, which differ in degrees of their noise resistance. More noise-resistant methods enable the reduction in radiation doses when conducting the examination of the patient. Hence, the development of noise-resistant methods is an important problem. This paper presents an algorithm for evaluating the quantitative characteristics of cerebral blood flow, based on the regularization using the projection onto a set of monotonic functions while minimizing the functional of total generalized variation (TGV). The proposed approach is tested on synthetic and real-world data. It yields better results than the singular value decomposition (SVD) method with Tikhonov regularization and methods of total variation (TV) minimization and TGV minimization.

We introduce a total variation (TV) regularization model for synthetic aperture radar (SAR) image despeckling. A dual-formulation-based adaptive TV (ATV) regularization method is applied to solve the TV regularization. The parameter adaptation of the TV regularization is performed based on the noise level estimated via wavelets. The TV-regularization-based image restoration model has a good performance in preserving image sharpness and edges while removing noises, and it is therefore effective for edge preserve SAR image despeckling. Experiments have been carried out using optical images contaminated with artificial speckles first and then SAR images. A despeckling evaluation index (DEI) is designed to assess the effectiveness of edge preserve despeckling on SAR images, which is based on the ratio of the standard deviations of two neighborhood areas of different sizes of a pixel. Experimental results show that the proposed ATV method can effectively suppress SAR image speckles without compromising the edge sharpness of image features according to both subjective visual assessment of image quality and objective evaluation using DEI.

This paper aims to introduce a convex prior based on gradient-domain image decomposition (GID) for image recovery. As a useful class of convex priors, total variation (TV) and its variants have been widely investigated. Among them, total generalized variation (TGV) is paid much attention recently, because it provides rich visual quality around edges and gradation regions in recovered images. This paper gives a general perspective: the TGV can be regarded as a GID and thus can be extended to more general formulation. Specifically, by introducing some priors promoting desired properties on gradient components which cannot be treated efficiently by the TGV, the restored image would attain better visual quality. As a practical instance, we incorporate block nuclear norm (BNN), which can characterize local low rankness, into the GID framework to keep the visual quality of well-patterned textures. Consequently, the GID provides better subjective visual quality and less reconstruction error in missing pixel recovery, than the TGV-based and the TV/BNN-based regularization methods.

We propose an adaptive total generalized variation (TGV) based model, aiming at achieving a balance between edge preservation and region smoothness for image denoising. The variable splitting (VS) and the classical augmented Lagrangian method (ALM) are used to solve the proposed model. With the proposed adaptive model and ALM, the regularization parameter, which balances the data fidelity and the regularizer, is refreshed with a closed form in each iterate, and the image denoising can be accomplished without manual interference. Numerical results indicate that our method is effective in staircasing effect suppression and holds superiority over some other state‐of‐the‐art methods both in quantitative and in qualitative assessment.

Total variation (TV) regularization can very well remove noise and simultaneously preserve the sharp edges. But it has the drawback of the contrast loss in the restoration. In this paper, we first theoretically analyze the loss of contrast in the original TV regularization model, and then propose a forward-backward diffusion model in the framework of total variation, which can effectively preserve the edges and contrast in TV image denoising. A backward diffusion term based on a nonconvex and monotony decrease potential function is introduced in the TV energy, resulting in a forward-backward diffusion. In order to finely control the strength of the forward and backward diffusion, and separately design the efficient algorithm to numerically implement the forward and backward diffusion, we propose a two-step splitting method to iteratively solve the proposed model. We adopt the efficient projection algorithm in the dual framework to solve the forward diffusion in the first step, and then use the simple finite differences scheme to solve the backward diffusion to compensate the loss of contrast occurred in the previous step. At last, we test the models on both synthetic and real images. Compared with the classical TV, forward and backward diffusion (FBD), two-step methods (TSM), and TV-FF models, our model has the better performance in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity (MSSIM) indexes.

It is required that input features are represented as vectors or scalars in machine learning for classification, e.g. support vector machine (SVM). However, real world data such as 2D images is naturally represented as matrices or tensors with higher dimensions. Thus, structural information of the data whose dimensions are more than two is not successfully considered. One typical structural information which is useful for the classification task is the spatial relationship of the nearby data points. In this paper, to leverage this kind of structural information, we propose a novel classification method which combines total variational (TV) regularization with SVM, called TV-SVM. Since TV achieves a local smoothing property by penalizing the local discontinuity of data, TV-SVM preserves better local structure than the original SVM due to TV regularization. We solve the objective function of TV-SVM via the alternating direction method of multipliers (ADMM) algorithm. Experimental results on image classification show that TV-SVM is competitive to the state-of-the-art learning method in both classification accuracy and computational complexity.

We propose a novel total variation based image restoration method that combines first and second order total variation functionals in a structurally adaptive manner. Specifically, as opposed to recent practical combined order methods, which choose the relative weight between the functionals of different derivative orders as a global scalar parameter, we construct a functional that uses a spatially varying relative weight between functionals of different orders so as to adapt to the local image structure. Further, as opposed to known combined order methods which leave the relative weight as a user parameter, the proposed method automatically computes the relative weight using a novel multi-resolution based formulation. The proposed spatially adaptive combined order TV (SAMTV) performs better than the state of art total generalized variation (TGV) method in experiments.