TV Regularization Research Articles

Hyperspectral image denoising by total variation-regularized bilinear factorization

Hyperspectral image (HSI) denoising is a prevalent research topic in the remote sensing area. In general, HSIs are inevitably impaired by different types of noise during the data acquisition. To fully characterize the underlying structures of clean HSI and remove mixed noises, we introduce a novel HSI denoising method named total variation-regularized bilinear factorization (BFTV) model. Specifically, we first utilize the bilinear factorization term to explore the globally low-rank structure of the clean HSI and suppress a certain degree of Gaussian noise, so as to make BFTV free to the singular value decomposition. Then the l1-norm is applied to detect and separate the mixed sparse noise including impulse noise, deadlines, and stripes. Besides, the TV regularization is introduced to describe the locally piecewise smoothness property of the clean HSI both in spatial and spectral domains. To solve this optimization problem, we devise an effective algorithm based on the augmented Lagrange multiplier method. Numerical experiments on five different kinds of mixed noise scenarios and one real world data have tested and demonstrated the superior denoising power of the proposed BFTV model compared with three state-of-the-art low-rank-based approaches.

Improved Collaborative Non-Negative Matrix Factorization and Total Variation for Hyperspectral Unmixing

Hyperspectral unmixing (HSU) is an important technique of remote sensing, which estimates the fractional abundances and the mixing matrix of endmembers in each mixed pixel from the hyperspectral image. Over the last years, the linear spectral unmixing problem has been approached as the sparse regression by different algorithms. Nevertheless, the huge solution space for individual pixels makes it difficult to search for the optimal solution in some HSU algorithms. Besides, the mixing relationship between adjacent pixels is not fully utilized as well. To better handle the huge solution space problem and explore the adjacent relationship, this article presents an improved collaborative non-negative matrix factorization and total variation algorithm (ICoNMF-TV) for HSU. The main contributions of this article are threefold: 1) a new framework named ICoNMF-TV based on non-negative matrix factorization method and TV regularization is developed to improve the performance of HSU algorithms; 2) unmixing efficiency is apparently improved; and 3) the robustness is enhanced. Experiment results on simulation dataset and real dataset demonstrate the proposed algorithm outperforms most of the similar sparse regression algorithms.

Multi-Frame Depth Super-Resolution for ToF Sensor With Total Variation Regularized L1 Function

In this paper, we propose a multi-frame depth super-resolution (SR) method based on L 1 data fidelity with the total variation regularization (TV-L 1 ) model. The majority of time-of-flight (ToF) sensors exhibit limited spatial resolution compared to RGB sensors and the improvement of the depth image resolution is an inherently ill-posed problem. To overcome this under-determined problem, the solution space is limited by the regularization term through prior knowledge and the data fidelity term using statistical information of the noise. Firstly, the statistical characteristics of ToF depth images are analyzed to specify the appropriate observation model. Thereafter, the objective function for multi-frame depth SR based on the TV-L 1 model is designed by considering the prior knowledge of the depth images. This approach enables the sharpness of the edges to be preserved and the noise amplification to be suppressed simultaneously. Furthermore, an efficient solver based on half-quadratic splitting is proposed. The algorithm minimizes the objective function for the multi-frame SR problem consisting of the TV regularization term and L 1 data fidelity term. The proposed method is verified on a synthetic dataset and real-world data acquired from a ToF sensor. The experimental results demonstrate that the proposed method can substantially reconstruct high-resolution depth images in terms of preserving sharp depth discontinuities, without any obvious artifacts, and can increase robustness to noise.

Sparse Unmixing for Hyperspectral Image with Nonlocal Low-Rank Prior

Hyperspectral unmixing is a key preprocessing technique for hyperspectral image analysis. To further improve the unmixing performance, in this paper, a nonlocal low-rank prior associated with spatial smoothness and spectral collaborative sparsity are integrated together for unmixing the hyperspectral data. The proposed method is based on a fact that hyperspectral images have self-similarity in nonlocal sense and smoothness in local sense. To explore the spatial self-similarity, nonlocal cubic patches are grouped together to compose a low-rank matrix. Then, based on the linear mixed model framework, the nuclear norm is constrained to the abundance matrix of these similar patches to enforce low-rank property. In addition, the local spatial information and spectral characteristic are also taken into account by introducing TV regularization and collaborative sparse terms, respectively. Finally, the results of the experiments on two simulated data sets and two real data sets show that the proposed algorithm produces better performance than other state-of-the-art algorithms.

ACQUIRE: an inexact iteratively reweighted norm approach for TV-based Poisson image restoration

We propose a method, called ACQUIRE, for the solution of constrained optimization problems modeling the restoration of images corrupted by Poisson noise. The objective function is the sum of a generalized Kullback–Leibler divergence term and a TV regularizer, subject to nonnegativity and possibly other constraints, such as flux conservation. ACQUIRE is a line-search method that considers a smoothed version of TV, based on a Huber-like function, and computes the search directions by minimizing quadratic approximations of the problem, built by exploiting some second-order information. A classical second-order Taylor approximation is used for the Kullback–Leibler term and an iteratively reweighted norm approach for the smoothed TV term. We prove that the sequence generated by the method has a subsequence converging to a minimizer of the smoothed problem and any limit point is a minimizer. Furthermore, if the problem is strictly convex, the whole sequence is convergent. We note that convergence is achieved without requiring the exact minimization of the quadratic subproblems; low accuracy in this minimization can be used in practice, as shown by numerical results. Experiments on reference test problems show that our method is competitive with well-established methods for TV-based Poisson image restoration, in terms of both computational efficiency and image quality.

A generalized matrix Krylov subspace method for TV regularization

This paper presents efficient algorithms to solve both TV/L1 and TV/L2 models of images contaminated by blur and noise. The unconstrained structure of the problems suggests that one can solve a constrained optimization problem by transforming the original unconstrained minimization problem to an equivalent constrained minimization one. An augmented Lagrangian method is developed to handle the constraints when the model is given with matrix variables, and an alternating direction method (ADM) is used to iteratively find solutions of the subproblems. The solutions of some subproblems are belonging to subspaces generated by application of successive orthogonal projections onto a class of generalized matrix Krylov subspaces of increasing dimension. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed algorithms.

ADMM-based multiparameter wavefield reconstruction inversion in VTI acoustic media with TV regularization

SUMMARYFull waveform inversion (FWI) is a nonlinear waveform matching procedure, which suffers from cycle skipping when the initial model is not kinematically accurate enough. To mitigate cycle skipping, wavefield reconstruction inversion (WRI) extends the inversion search space by computing wavefields with a relaxation of the wave equation in order to fit the data from the first iteration. Then, the subsurface parameters are updated by minimizing the source residuals the relaxation generated. Capitalizing on the wave-equation bilinearity, performing wavefield reconstruction and parameter estimation in alternating mode decomposes WRI into two linear subproblems, which can be solved efficiently with the alternating-direction method of multiplier (ADMM), leading to the so-called iteratively refined WRI (IR–WRI). Moreover, ADMM provides a suitable framework to implement bound constraints and different types of regularizations and their mixture in IR–WRI. Here, IR–WRI is extended to multiparameter reconstruction for vertical transverse isotropic (VTI) acoustic media. To achieve this goal, we first propose different forms of bilinear VTI acoustic wave equation. We develop more specifically IR–WRI for the one that relies on a parametrization involving vertical wave speed and Thomsen’s parameters δ and ϵ. With a toy numerical example, we first show that the radiation patterns of the virtual sources generate similar wavenumber filtering and parameter cross-talks in classical FWI and IR–WRI. Bound constraints and TV regularization in IR–WRI fully remove these undesired effects for an idealized piecewise constant target. We show with a more realistic long-offset case study representative of the North Sea that anisotropic IR–WRI successfully reconstruct the vertical wave speed starting from a laterally homogeneous model and update the long wavelengths of the starting ϵ model, while a smooth δ model is used as a passive background model. VTI acoustic IR–WRI can be alternatively performed with subsurface parametrizations involving stiffness or compliance coefficients or normal moveout velocities and η parameter (or horizontal velocity).

Hybrid non-convex second-order total variation with applications to non-blind image deblurring

Total variation is a very popular method for image restoration, yet it produces undesirable staircase artifacts. In this paper, a combined second-order non-convex total variation with overlapping group sparse regularizer for staircase artifact removal is proposed. The non-convex higher-order TV regularizer is introduced to model the observation that the staircase effect must be appropriately smoothen out in the restored image while preserving its edges. However, using the non-convex higher-order TV alone tends to smoothen the image while amplifying speckle artifacts. To deal with this, the overlapping group sparse regularizer is added to balance the effects produced by the non-convex higher-order TV regularizer. An efficient re-weighted $$\ell _1$$ alternating direction method is formulated to solve the corresponding iterative scheme. Comparative analysis with three algorithms, namely overlapping group sparse total variation, total generalized variation and the fast non-smooth non-convex method, with two different blur kernels is carried out. The results from peak signal-to-noise ratio and structural similarity index measure show the effectiveness of our proposed method when compared to the mentioned algorithms.

B-Spline-Based Exact Discretization of Continuous-Domain Inverse Problems With Generalized TV Regularization

We study continuous-domain linear inverse problems with generalized total-variation (gTV) regularization, expressed in terms of a regularization operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer jumps than the number of measurements. Moreover, the type of spline solely depends on L (L-splines) and is independent of the measurements. The continuous-domain inverse problem can be recast in an exact way as a finite-dimensional problem by restricting the search space to splines with knots on a uniform finite grid. However, expressing the L-spline coefficients in the dictionary basis of the Green’s function of L is ill-suited for practical problems due to its infinite support. Instead, we propose to formulate the problem in the B-spline dictionary basis, which leads to better-conditioned problems. As we make the grid finer, we show that a solution of the continuous-domain problem can be approached arbitrarily closely with functions of this search space. This result motivates our proposed multiresolution algorithm, which computes sparse solutions of our inverse problem. We demonstrate that this algorithm is computationally feasible for 1D signals when L is an ordinary differential operator.

Adaptive Gaussian Weighted Laplace Prior Regularization Enables Accurate Morphological Reconstruction in Fluorescence Molecular Tomography.

Fluorescence molecular tomography (FMT), as a powerful imaging technique in preclinical research, can offer the three-dimensional distribution of biomarkers by detecting the fluorescently labelled probe noninvasively. However, because of the light scattering effect and the ill-pose of inverse problem, it is challenging to develop an efficient reconstruction method, which can provide accurate location and morphology of the fluorescence distribution. In this research, we proposed a novel adaptive Gaussian weighted Laplace prior (AGWLP) regularization method, which assumed the variance of fluorescence intensity between any two voxels had a non-linear correlation with their Gaussian distance. It utilized an adaptive Gaussian kernel parameter strategy to achieve accurate morphological reconstructions in FMT. To evaluate the performance of the AGWLP method, we conducted numerical simulation and in vivo experiments. The results were compared with fast iterative shrinkage (FIS) thresholding method, split Bregman-resolved TV (SBRTV) regularization method, and Gaussian weighted Laplace prior (GWLP) regularization method. We validated in vivo imaging results against planar fluorescence images of frozen sections. The results demonstrated that the AGWLP method achieved superior performance in both location and shape recovery of fluorescence distribution. This enabled FMT more suitable and practical for in vivo visualization of biomarkers.

One-step deconvolution for multi-angle TIRF microscopy with enhanced resolution.

Total internal reflection fluorescence microscopy (TIRF microscopy) uses a rapid decay of evanescent waves to excite fluorophores within several hundred nanometers (nm) beneath the plasma membrane, which can effectively suppress excitation of fluorescence signals in the deep layers. From image stacks obtained with a plurality of different incident angles, a three-dimensional spatial structure of the observed sample can be reconstructed by a Multi-Angle-TIRF (MA-TIRF) algorithm that provides an axial resolution of ~50 nm. Taking into account the point spread function (PSF) of the TIRF microscopes, we further increase its lateral resolution by introducing a fast deconvolution algorithm into the reconstruction of MA-TIRF data (DMA-TIRF), which is approached in just one step of minimizing the reconstruction function. We also introduce a TV regularization term in the deconvolution algorithm to suppress artifacts induced by the excessive noise. Therefore, based on the hardware of existing MA-TIRF microscopes, the proposed DMA-TIRF algorithm has achieved lateral and axial resolutions of ~200 and ~50 nm, respectively.

Total Variation Regularized Collaborative Representation Clustering With a Locally Adaptive Dictionary for Hyperspectral Imagery

Clustering is a very challenging task for hyperspectral imagery (HSI) because of the complex spectral–spatial structures found in such data. Recently, the sparse recovery-based approaches have been introduced to deal with hyperspectral clustering, and have achieved state-of-the-art performances. Several recent works have shown that it is the collaborative representation mechanism over all the dictionary atoms, rather than the sparse constraint that determines the recognition performance. Based on this fact, in this paper, we focus on the working mechanism of collaborative representation to explore its potential in HSI clustering. However, directly introducing collaborative representation clustering (CRC) to HSIs results in several problems, i.e., the high redundancy of the global dictionary atoms and the absence of spatial information, which greatly limit the clustering performance. In view of this, we propose a novel total variation regularized CRC with a locally adaptive dictionary (TV-CRC-LAD) algorithm for HSI. First, the LAD construction strategy is introduced instead of the global dictionary to relieve the high redundancy and the interference of unrelated atoms in the representation process, to more precisely represent each pixel only with the highly correlated atoms. Second, TV regularization is integrated to better account for the rich spatial-contextual information and promotes the piecewise smoothness of the HSI clustering result. The proposed algorithm was tested on three widely used hyperspectral data sets, and the experimental results clearly illustrate that the proposed algorithm outperforms the corresponding sparsity-based clustering methods and the other state-of-the-art methods.

Ordered Subsets EM algorithm for PET Image Reconstruction by use of Dictionary Learning and TV Regularization

Ordered Subsets EM algorithm for PET Image Reconstruction by use of Dictionary Learning and TV Regularization

Retraction notice.

Reducing radiation dose while maintaining the quality of the reconstructed images is a major challenge in the computed tomography (CT) community. In light of the non-stationary Gaussian noise distribution, we developed a model that incorporates a noise-level weighted total variation (NWTV) regularization term for denoising the projection data. Contrary to the well-known edge-weighted total variation method, which aims for better edge preserving, the proposed NWTV tries to adapt the regularization with the spatially varying noise levels. Experiments on simulated data as well as the real imaging data suggest that the proposed NWTV regularization could achieve quite competitive results. For sinograms with sharp edges, the NWTV could do a better job at balancing noise reduction and edge preserving, such that noise is removed in a more uniform manner. Another conclusion from our experiments is that the well-recognized stair-casing artifacts of TV regularization play little role in the reconstructed images when the NWTV method is applied to low-dose CT imaging data.

Edge-adaptive <inline-formula><tex-math id="M1">\begin{document}$ \ell_2 $\end{document}</tex-math></inline-formula> regularization image reconstruction from non-uniform Fourier data

Total variation regularization based on the l1 norm is ubiquitous in image reconstruction. However, the resulting reconstructions are not always as sparse in the edge domain as desired. Iteratively reweighted methods provide some improvement in accuracy, but at the cost of extended runtime. In this paper we examine these methods for the case of data acquired as non-uniform Fourier samples. We then develop a non-iterative weighted regularization method that uses a pre-processing edge detection to find exactly where the sparsity should be in the edge domain. We show that its performance in terms of both accuracy and speed has the potential to outperform reweighted TV regularization methods.

Total generalized variation regularization in data assimilation for Burgers' equation

We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

Robustly reconstructing magnetic resonance images via structure decomposition

In magnetic resonance (MR) imaging, for highly under-sampled k-space data, it is typically difficult to reconstruct images and preserve their original texture simultaneously. The high-degree total variation (HDTV) regularization handles staircase effects but still blurs textures. On the other hand, the non-local TV (NLTV) regularization can preserve textures, but will introduce additional artifacts for highly-noised images. In this paper, we propose a reconstruction model derived from HDTV and NLTV for robust MRI reconstruction. First, an MR image is decomposed into a smooth component and a texture component. Second, for the smooth component with sharp edges, isotropic second-order TV is used to reduce staircase effects. For the texture component with piecewise constant background, NLTV and contourlet-based sparsity regularizations are employed to recover textures. The piecewise constant background in the texture component contributes to accurately detect non-local similar image patches and avoid artifacts introduced by NLTV. Finally, the proposed reconstruction model is solved through an alternating minimization scheme. The experimental results demonstrate that the proposed reconstruction model can effectively achieve satisfied quality of reconstruction for highly under-sampled k-space data.

Regularized inversion of amplitude-versus-incidence angle (AVA) based on a piecewise-smooth model

Different from the stacked seismic data, pre-stack data includes abundant information about shear wave and density. Through inversing the shear wave and density information from the pre-stack data, we can determine oil-bearing properties from different incident angles. The state-of-the-art inversion methods obtain either low vertical resolution or lateral discontinuities. However, the practical reservoir generally has sharp discontinuities between different layers in vertically direction and is horizontally smooth. Towards obtaining the practical model, we present an inversion method based on the regularized amplitude-versus-incidence angle (AVA) data to estimate the piecewise-smooth model from pre-stack seismic data. This method considers subsurface stratum as a combination of two parts: a piecewise smooth part and a constant part. To fix the ill-posedness in the inversion, we adopt four terms to define the AVA inversion misfit function: the data misfit itself, a total variation regularization term acting as a sparsing operator for the piecewise constant part, a Tikhonov regularization term acting as a smoothing operator for the smooth part, and the last term to smoothly incorporate a priori information for constraining the magnitude of the estimated model. The proposed method not only can incorporate structure information and a priori model constraint, but also is able to derive into a convex objective function that can be easily minimized using iterative approach. Compared with inversion results of TV and Tikhonov regularization methods, the inverted P-wave velocity, S-wave velocity and density of the proposed method can better delineate the piecewise-smooth characteristic of strata.

Hyperspectral Unmixing Using Sparsity-Constrained Deep Nonnegative Matrix Factorization With Total Variation

Hyperspectral unmixing is an important processing step for many hyperspectral applications, mainly including: 1) estimation of pure spectral signatures (endmembers) and 2) estimation of the abundance of each endmember in each pixel of the image. In recent years, nonnegative matrix factorization (NMF) has been highly attractive for this purpose due to the nonnegativity constraint that is often imposed in the abundance estimation step. However, most of the existing NMF-based methods only consider the information in a single layer while neglecting the hierarchical features with hidden information. To alleviate such limitation, in this paper, we propose a new sparsity-constrained deep NMF with total variation (SDNMF-TV) technique for hyperspectral unmixing. First, by adopting the concept of deep learning, the NMF algorithm is extended to deep NMF model. The proposed model consists of pretraining stage and fine-tuning stage , where the former pretrains all factors layer by layer and the latter is used to reduce the total reconstruction error. Second, in order to exploit adequately the spectral and spatial information included in the original hyperspectral image, we enforce two constraints on the abundance matrix. Specifically, the $L_{1/2}$ constraint is adopted, since the distribution of each endmember is sparse in the 2-D space. The TV regularizer is further introduced to promote piecewise smoothness in abundance maps. For the optimization of the proposed model, multiplicative update rules are derived using the gradient descent method. The effectiveness and superiority of the SDNMF-TV algorithm are demonstrated by comparing with other unmixing methods on both synthetic and real data sets.

A fast total variation regularization algorithm for 2D piecewise constant radially symmetric functions

In this paper, we considered total variation regularization (TV regularization) for 2D radially symmetric piecewise constant (RSPC) functions. We obtained a system of equations solving the direct variational problem with the subgradient method. Using the system, we propose a Condat’s type algorithm for computation of an extremal function.