Total Variation Minimization Problem Research Articles

A Uniform Preconditioner for a Newton Algorithm for Total Variation Minimization and Minimum-Surface Problems

A Uniform Preconditioner for a Newton Algorithm for Total Variation Minimization and Minimum-Surface Problems

Strip-Map SAR Image Formulation Based on the Modified Alternating Split Bregman Method

Conventional compressive sensing (CS)-based imaging methods allow images to be reconstructed from a small amount of data, while they suffer from high computational burden even for a moderate scene. To address this problem, this paper presents a novel two-dimensional (2D) CS imaging algorithm for strip-map synthetic aperture radars (SARs) with zero squint angle. By introducing a 2D separable formulation to model the physical procedure of the SAR imaging, we separate the large measurement matrix into two small ones, and then the induced algorithm can deal with 2D signal directly instead of converting it into 1D vector. As a result, the computational load can be reduced significantly. Furthermore, thanks to its superior performance in maintaining contour information, the gradient space of the SAR image is exploited and the total variation (TV) constraint is incorporated to improve resolution performance. Due to the non-differentiable property of the TV regularizer, it is difficult to directly solve the induced TV regularization problem. To overcome this problem, an improved split Bregman method is presented by formulating the TV minimization problem into a sequence of unconstrained optimization problem and Bregman updates. It yields an accurate and simple solution. Finally, the synthesis and real experiment results demonstrate that the proposed algorithm remains competitive in terms of high resolution and high computational efficiency.

An efficient multi-grid method for TV minimization problems

<p style='text-indent:20px;'>We propose an efficient multi-grid domain decomposition method for solving the total variation (TV) minimization problems. Our multi-grid scheme is developed based on the piecewise constant function spanned subspace correction rather than the piecewise linear one in [<xref ref-type="bibr" rid="b17">17</xref>], which ensures the calculation of the TV term only occurs on the boundaries of the support sets. Besides, the domain decomposition method is implemented on each layer to enable parallel computation. Comprehensive comparison results are presented to demonstrate the improvement in CPU time and image quality of the proposed method on medium and large-scale image denoising and reconstruction problems.</p>

Research on ADMM Reconstruction Algorithm of Photoacoustic Tomography With Limited Sampling Data

Photoacoustic imaging is a new non-destructive biomedical imaging method. When limited independent data is available, the restoration of the initial pressure rise distribution is often an ill-posed problem. In this paper, based on the study of photoacoustic effects, the sparse prior information of photoacoustic images is integrated into the reconstruction process by using the compressed sensing (CS) theory and the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> norm optimization technique, combining the augmented Langrange weighting of the alternating direction method of multipliers (ADMM) with the total variation (TV) minimization problem, and the reconstruction artifacts are effectively eliminated. The simulation data from the real numerical model show that compared with the common time reversal algorithm, interpolation algorithm and truncated back projection algorithm, the total variational regularization method based on ADMM can effectively improve the quality of reconstructed images under the condition of limited viewing angles and incomplete projection data.

Sparse deconvolution via off-grid T.V minimization

We present a grid-less solution to the total variation minimization approach to the sparse deconvolution problem. Unlike the existing methods [1, 2], our algorithm works directly with time domain data, and does not suffer from the consequences of leakage errors incurred while transforming the time domain measurements to frequency domain. The total variation minimization problem requires us to optimize the total variation of a function. This is an infinite dimensional problem. To deal with the underlying infinite dimensionality, we consider its finite dimensional semi-infinite duals with point-wise constraints. We provide a finite parameterization to handle the point-wise constraints in a computationally tractable manner. This parameterization exploits the underlying bandlimitedness of the convolution kernel by employing Prolate Spheroidal Wave Functions. Consequently, we are able to reduce the total variation minimization problem into a semidefinite program, which can be solved in polynomial time. We demonstrate its utility via numerical simulation studies as well as real world signal reconstruction examples.

A primal-dual multiplier method for total variation image restoration

In this paper, we propose a novel primal-dual multiplier method for total variation minimization problems. The new idea combining Lagrange dual and total variation dual is presented. Convergence of the proposed algorithms is easily obtained from the proximal point algorithm frame and numerical results including image deblurring, parallel magnetic resonance imaging and image inpainting demonstrate the effectiveness of the new algorithms.

Fusing Infrared and Visible Images of Different Resolutions via Total Variation Model.

In infrared and visible image fusion, existing methods typically have a prerequisite that the source images share the same resolution. However, due to limitations of hardware devices and application environments, infrared images constantly suffer from markedly lower resolution compared with the corresponding visible images. In this case, current fusion methods inevitably cause texture information loss in visible images or blur thermal radiation information in infrared images. Moreover, the principle of existing fusion rules typically focuses on preserving texture details in source images, which may be inappropriate for fusing infrared thermal radiation information because it is characterized by pixel intensities, possibly neglecting the prominence of targets in fused images. Faced with such difficulties and challenges, we propose a novel method to fuse infrared and visible images of different resolutions and generate high-resolution resulting images to obtain clear and accurate fused images. Specifically, the fusion problem is formulated as a total variation (TV) minimization problem. The data fidelity term constrains the pixel intensity similarity of the downsampled fused image with respect to the infrared image, and the regularization term compels the gradient similarity of the fused image with respect to the visible image. The fast iterative shrinkage-thresholding algorithm (FISTA) framework is applied to improve the convergence rate. Our resulting fused images are similar to super-resolved infrared images, which are sharpened by the texture information from visible images. Advantages and innovations of our method are demonstrated by the qualitative and quantitative comparisons with six state-of-the-art methods on publicly available datasets.

Adaptive image fusion algorithm based on human visual system guided gradient transfer and total variation minimization

To effectively explore image content for image fusion and generate a full end-to-end image fusion pipeline, a parameter adaption algorithm based on gradient transfer and total variation (TV) minimization is proposed, and the fusion problem is transformed into a human visual system guided TV minimization problem, where the data fidelity term maintains the structure edge features of the image, and the regularization term preserves the gradient variation texture detail information. Then, the fusion problem is cast into the optimization of minimum energy functional model. To facilitate choice of regularization parameter, a modified fixed-point iterative method is employed to solve the optimization. Extensive experiments show that this model can obtain both rich detail and spectral information; adjusting the parameters of the proposed adaptive selection strategy can effectively improve the fusion effect while making the image fusion process work as an end-to-end pipeline. Compared to eight state-of-the-art fusion methods, the experimental results demonstrate the effectiveness of the proposed method.

MPTV: Matching Pursuit-Based Total Variation Minimization for Image Deconvolution.

Total variation (TV) regularization has proven effective for a range of computer vision tasks through its preferential weighting of sharp image edges. Existing TV-based methods, however, often suffer from the over-smoothing issue and solution bias caused by the homogeneous penalization. In this paper, we consider addressing these issues by applying inhomogeneous regularization on different image components. We formulate the inhomogeneous TV minimization problem as a convex quadratic constrained linear programming problem. Relying on this new model, we propose a matching pursuit-based total variation minimization method (MPTV), specifically for image deconvolution. The proposed MPTV method is essentially a cutting-plane method that iteratively activates a subset of nonzero image gradients and then solves a subproblem focusing on those activated gradients only. Compared with existing methods, the MPTV is less sensitive to the choice of the trade-off parameter between data fitting and regularization. Moreover, the inhomogeneity of MPTV alleviates the over-smoothing and ringing artifacts and improves the robustness to errors in blur kernel. Extensive experiments on different tasks demonstrate the superiority of the proposed method over the current state of the art.

An extended variational image decomposition model for color image enhancement

Abstract Variational image decomposition model has been widely used in computer vision and computational photography to decompose an image into a luminance layer and a reflectance layer. Existing method perform decomposition either on the intensity channel of a color image or on each of the RGB channels separately. However, these methods fail to exploit effectively the correlation among color channels, and will lose much color edges and textures during the decomposition process. This paper presents an extended variational image decomposition model, which works directly on the color image and decomposes it into a luminance layer, a reflectance layer and a color layer by solving a total variation minimization problem. The luminance layer is piecewise smooth containing the general object appearance, the reflectance layer is piecewise constant representing some structure information. The color layer accounts for distinct color structure in each color channel and reveals the color variation of the image. By manipulating the three layers, the proposed decomposition model can be applied to a variety of color image decomposition applications such as high dynamic range tone mapping and contrast enhancement. Our experimental results validate the highly competitive performance of the proposed method in comparison with state-of-the-arts.

Domain Decomposition Methods Using Dual Conversion for the Total Variation Minimization with $$L^1$$ L 1 Fidelity Term

Nowadays, as large scale images become available, the necessity of parallel algorithms for image processing has been arisen. In this paper, we propose domain decomposition methods as parallel solvers for solving total variation minimization problems with $$L^1$$ fidelity term. The image domain is decomposed into rectangular subdomains, where the local total variation problems are solved. We introduce the notion of dual conversion, which generalizes the framework of Chambolle–Pock primal-dual algorithm (J Math Imaging Vis 40:120–145, 2011). By the dual conversion, the TV- $$L^1$$ model is transformed into an equivalent saddle point problem which has a natural parallel structure. The primal problem of the resulting saddle point problem is decoupled in the sense that each local problem can be solved independently. Convergence analysis of the proposed algorithms is provided. We apply the proposed algorithms for image denoising, inpainting, and deblurring problems. Our numerical results ensure that the proposed algorithms have good performance as parallel solvers.

A simple primal-dual algorithm for nuclear norm and total variation regularization

Abstract We propose a simple primal-dual method for nuclear norm plus total variation minimization problems. A predictor-corrector scheme to the dual variable is used in our algorithm. Convergence of the method is proved and convergence rate which is O(1/N) in the ergodic sense is also discussed, where N denotes the iteration number. Numerical results including tensor completion, parallel magnetic resonance imaging and dynamic magnetic resonance imaging demonstrate the efficiency of the new algorithm.

A low-rank approach to off-the-grid sparse deconvolution

We propose a new solver for the sparse spikes deconvolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite (SDP) relaxations of the total variation (the total mass of the measure) minimization problem. The direct resolution of this SDP is however intractable for large scale settings, since the problem size grows as fc2d where fc is the cutoff frequency of the filter. Our first contribution introduces a penalized formulation of this semidefinite lifting, which has low-rank solutions. Our second contribution is a conditional gradient optimization scheme with non-convex updates. This algorithm leverages both the low-rank and the convolutive structure of the problem, resulting in an O(fcd log fc) complexity per iteration. Numerical simulations are promising and show that the algorithm converges in exactly k steps, k being the number of Diracs composing the solution.

Graph Signal Recovery via Primal-Dual Algorithms for Total Variation Minimization

We consider the problem of recovering a smooth graph signal from noisy samples taken on a subset of graph nodes. The smoothness of the graph signal is quantified in terms of total variation. We formulate the signal recovery task as a convex optimization problem that minimizes the total variation of the graph signal while controlling its global or node-wise empirical error. We propose a first-order primal-dual algorithm to solve these total variation minimization problems. A distributed implementation of the algorithm is devised to handle large-dimensional applications efficiently. We use synthetic and real-world data to extensively compare the performance of our approach with state-of-the-art methods.

Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees

Image restoration is one of the essential tasks in image processing. In order to restore images from blurs and noise while also preserving their edges, one often applies total variation (TV) minimization. Cauchy noise, which frequently appears in engineering applications, is a kind of impulsive and non-Gaussian noise. Removing Cauchy noise can be achieved by solving a nonconvex TV minimization problem, which is difficult due to its nonconvexity and nonsmoothness. In this paper, we adapt recent results in the literature and develop a specific alternating direction method of multiplier to solve this problem. Theoretically, we establish the convergence of our method to a stationary point. Experimental results demonstrate that the proposed method is competitive with other methods in visual and quantitative measures. In particular, our method achieves higher PSNRs for 0.5 dB on average.

Total Variation Based Parameter-Free Model for Impulse Noise Removal

AbstractWe propose a new two-phase method for reconstruction of blurred images corrupted by impulse noise. In the first phase, we use a noise detector to identify the pixels that are contaminated by noise, and then, in the second phase, we reconstruct the noisy pixels by solving an equality constrained total variation minimization problem that preserves the exact values of the noise-free pixels. For images that are only corrupted by impulse noise (i.e., not blurred) we apply the semismooth Newton's method to a reduced problem, and if the images are also blurred, we solve the equality constrained reconstruction problem using a first-order primal-dual algorithm. The proposed model improves the computational efficiency (in the denoising case) and has the advantage of being regularization parameter-free. Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

An Interior-Point Method for Modified Total Variation Exploiting Transform-Domain Sparsity

The total variation (TV) minimization can be utilized in a compressive sensing framework to recover a signal from a small number of measurements by searching for a signal with a sparse gradient. However, many natural signals of interest, such as natural images, generally have sparse representations in known transforms. Hence, the performance of the signal reconstruction procedure can be improved by also taking into account this transform-domain sparsity of the signal. Thus, the TV minimization problem can be modified by introducing an $l_1$-norm penalty term. The $l_1$-regularized TV minimization problem searches for a signal with a sparse gradient and a sparse representation in the given transform. The main contribution of this paper is the derivation of a customized interior-point method for solving the $l_1$-regularized TV minimization problem that computes the search direction of the Newton method efficiently by exploiting the specific structure of the Hessian.

A Coordinate Descent Method for Total Variation Minimization

Total variation (TV) is a well-known image model with extensive applications in various images and vision tasks, for example, denoising, deblurring, superresolution, inpainting, and compressed sensing. In this paper, we systematically study the coordinate descent (CoD) method for solving general total variation (TV) minimization problems. Based on multidirectional gradients representation, the proposed CoD method provides a unified solution for both anisotropic and isotropic TV-based denoising (CoDenoise). With sequential sweeping and small random perturbations, CoDenoise is efficient in denoising and empirically converges to optimal solution. Moreover, CoDenoise also delivers new perspective on understanding recursive weighted median filtering. By incorporating with the Augmented Lagrangian Method (ALM), CoD was further extended to TV-based image deblurring (ALMCD). The results on denoising and deblurring validate the efficiency and effectiveness of the CoD-based methods.

Primal Domain Decomposition Methods for the Total Variation Minimization, Based on Dual Decomposition

We propose nonoverlapping domain decomposition methods for solving the total variation minimization problem. We decompose the domain of the dual problem into nonoverlapping rectangular subdomains, where local total variation problems are solved. We convert the local dual problems into the equivalent primal forms which reproduce the original problem at smaller dimensions. Sequential and parallel algorithms are presented. The convergence of both algorithms is analyzed and numerical results are presented.

Iterative finite element solution of a constrained total variation regularized model problem

The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the non-differentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.