Total Variation Regularization Problem Research Articles

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.

Error Estimations for Total Variation Type Regularization

This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed; the results of the corresponding convergence rate are also presented under certain mild conditions.

A unified approach for a 1D generalized total variation problem

We study a 1-dimensional discrete signal denoising problem that consists of minimizing a sum of separable convex fidelity terms and convex regularization terms, the latter penalize the differences of adjacent signal values. This problem generalizes the total variation regularization problem. We provide here a unified approach to solve the problem for general convex fidelity and regularization functions that is based on the Karush–Kuhn–Tucker optimality conditions. This approach is shown here to lead to a fast algorithm for the problem with general convex fidelity and regularization functions, and a faster algorithm if, in addition, the fidelity functions are differentiable and the regularization functions are strictly convex. Both algorithms achieve the best theoretical worst case complexity over existing algorithms for the classes of objective functions studied here. Also in practice, our C++ implementation of the method is considerably faster than popular C++ nonlinear optimization solvers for the problem.

Airborne Radar Super-Resolution Imaging Based on Fast Total Variation Method

Total variation (TV) is an effective super-resolution method to improve the azimuth resolution and preserve the contour information of the target in airborne radar imaging. However, the computational complexity is very high because of the matrix inversion, reaching O(N3). In this paper, a Gohberg–Semencul (GS) representation based fast TV (GSFTV) method is proposed to make up for the shortcoming. The proposed GSFTV method fist utilizes a one-dimensional TV norm as the regular term under regularization framework, which is conducive to achieve super-resolution while preserving the target contour. Then, aiming at the very high computational complexity caused by matrix inversion when minimizing the TV regularization problem, we use the low displacement rank feature of Toeplitz matrix to achieve fast inversion through GS representation. This reduces the computational complexity from O(N3) to O(N2), benefiting efficiency improvement for airborne radar imaging. Finally, the simulation and real data processing results demonstrate that the proposed GSFTV method can simultaneously improve the resolution and preserve the target contour. Moreover, the very high computational efficiency of the proposed GSFTV method is tested by hardware platform.

Learning needle tip localization from digital subtraction in 2D ultrasound.

This paper addresses localization of needles inserted both in-plane and out-of-plane in challenging ultrasound-guided interventions where the shaft and tip have low intensity. Our approach combines a novel digital subtraction scheme for enhancement of low-level intensity changes caused by tip movement in the ultrasound image and a state-of-the-art deep learning scheme for tip detection. As the needle tip moves through tissue, it causes subtle spatiotemporal variations in intensity. Relying on these intensity changes, we formulate a foreground detection scheme for enhancing the tip from consecutive ultrasound frames. The tip is augmented by solving a spatial total variation regularization problem using the split Bregman method. Lastly, we filter irrelevant motion events with a deep learning-based end-to-end data-driven method that models the appearance of the needle tip in ultrasound images, resulting in needle tip detection. The detection model is trained and evaluated on an extensive ex vivo dataset collected with 17G and 22G needles inserted in-plane and out-of-plane in bovine, porcine and chicken phantoms. We use 5000 images extracted from 20 video sequences for training and 1000 images from 10 sequences for validation. The overall framework is evaluated on 700 images from 20 sequences not used in training and validation, and achieves a tip localization error of 0.72 ± 0.04mm and an overall processing time of 0.094s per frame (~ 10 frames per second). The proposed method is faster and more accurate than state of the art and is resilient to spatiotemporal redundancies. The promising results demonstrate its potential for accurate needle localization in challenging ultrasound-guided interventions.

Performance of relaxed iterative methods for image deblurring problems

In this paper, we consider performance of relaxation iterative methods for four types of image deblurring problems with different regularization terms. We first study how to apply relaxation iterative methods efficiently to the Tikhonov regularization problems, and then we propose how to find good preconditioners and near optimal relaxation parameters which are essential factors for fast convergence rate and computational efficiency of relaxation iterative methods. We next study efficient applications of relaxation iterative methods to Split Bregman method and the fixed point method for solving the L1-norm or total variation regularization problems. Lastly, we provide numerical experiments for four types of image deblurring problems to evaluate the efficiency of relaxation iterative methods by comparing their performances with those of Krylov subspace iterative methods. Numerical experiments show that the proposed techniques for finding preconditioners and near optimal relaxation parameters of relaxation iterative methods work well for image deblurring problems. For the L1-norm and total variation regularization problems, Split Bregman and fixed point methods using relaxation iterative methods perform quite well in terms of both peak signal to noise ratio values and execution time as compared with those using Krylov subspace methods.

Using generalized cross validation to select regularization parameter for total variation regularization problems

The regularization approach is used widely in image restoration problems. The visual quality of the restored image depends highly on the regularization parameter. In this paper, we develop an automatic way to choose a good regularization parameter for total variation (TV) image restoration problems. It is based on the generalized cross validation (GCV) approach and hence no knowledge of noise variance is required. Due to the lack of the closed-form solution of the TV regularization problem, difficulty arises in finding the minimizer of the GCV function directly. We reformulate the TV regularization problem as a minimax problem and then apply a first-order primal-dual method to solve it. The primal subproblem is rearranged so that it becomes a special Tikhonov regularization problem for which the minimizer of the GCV function is readily computable. Hence we can determine the best regularization parameter in each iteration of the primal-dual method. The regularization parameter for the original TV regularization problem is then obtained by an averaging scheme. In essence, our method needs only to solve the TV regulation problem twice: one to determine the regularization parameter and one to restore the image with that parameter. Numerical results show that our method gives near optimal parameter, and excellent performance when compared with other state-of-the-art adaptive image restoration algorithms.

Bregman iterative algorithms for 2D geosounding inversion

Bregman iterative algorithms have been extensively used for and total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure.

High-accuracy total variation with application to compressed video sensing.

Numerous total variation (TV) regularizers, engaged in image restoration problem, encode the gradients by means of simple [-1, 1] finite-impulse-response (FIR) filter. Despite its low computational processing, this filter severely distorts signal's high-frequency components pertinent to edge/ discontinuous information and cause several deficiency issues known as texture and geometric loss. This paper addresses this problem by proposing an alternative model to the TV regularization problem via high-order accuracy differential FIR filters to preserve rapid transitions in signal recovery. A numerical encoding scheme is designed to extend the TV model into multidimensional representation (tensorial decomposition). We adopt this design to regulate the spatial and temporal redundancy in compressed video sensing problem to jointly recover frames from undersampled measurements. We then seek the solution via alternating direction methods of multipliers and find a unique solution to quadratic minimization step with capability of handling different boundary conditions. The resulting algorithm uses much lower sampling rate and highly outperforms alternative state-of-the-art methods. This is evaluated both in terms of restoration accuracy and visual quality of the recovered frames.

Micro-CT image reconstruction based on alternating direction augmented Lagrangian method and total variation

Micro-computed tomography (micro-CT) plays an important role in pre-clinical imaging. The radiation from micro-CT can result in excess radiation exposure to the specimen under test, hence the reduction of radiation from micro-CT is essential. The proposed research focused on analyzing and testing an alternating direction augmented Lagrangian (ADAL) algorithm to recover images from random projections using total variation (TV) regularization. The use of TV regularization in compressed sensing problems makes the recovered image quality sharper by preserving the edges or boundaries more accurately. In this work TV regularization problem is addressed by ADAL which is a variant of the classic augmented Lagrangian method for structured optimization. The per-iteration computational complexity of the algorithm is two fast Fourier transforms, two matrix vector multiplications and a linear time shrinkage operation. Comparison of experimental results indicate that the proposed algorithm is stable, efficient and competitive with the existing algorithms for solving TV regularization problems.

Total variation regularization for bioluminescence tomography with the split Bregman method

Regularization methods have been broadly applied to bioluminescence tomography (BLT) to obtain stable solutions, including l2 and l1 regularizations. However, l2 regularization can oversmooth reconstructed images and l1 regularization may sparsify the source distribution, which degrades image quality. In this paper, the use of total variation (TV) regularization in BLT is investigated. Since a nonnegativity constraint can lead to improved image quality, the nonnegative constraint should be considered in BLT. However, TV regularization with a nonnegativity constraint is extremely difficult to solve due to its nondifferentiability and nonlinearity. The aim of this work is to validate the split Bregman method to minimize the TV regularization problem with a nonnegativity constraint for BLT. The performance of split Bregman-resolved TV (SBRTV) based BLT reconstruction algorithm was verified with numerical and in vivo experiments. Experimental results demonstrate that the SBRTV regularization can provide better regularization quality over l2 and l1 regularizations.

Selection of regularization parameter in total variation image restoration

We consider and study total variation (TV) image restoration. In the literature there are several regularization parameter selection methods for Tikhonov regularization problems (e.g., the discrepancy principle and the generalized cross-validation method). However, to our knowledge, these selection methods have not been applied to TV regularization problems. The main aim of this paper is to develop a fast TV image restoration method with an automatic selection of the regularization parameter scheme to restore blurred and noisy images. The method exploits the generalized cross-validation (GCV) technique to determine inexpensively how much regularization to use in each restoration step. By updating the regularization parameter in each iteration, the restored image can be obtained. Our experimental results for testing different kinds of noise show that the visual quality and SNRs of images restored by the proposed method is promising. We also demonstrate that the method is efficient, as it can restore images of size 256 x 256 in approximately 20 s in the MATLAB computing environment.

Structural Properties of Solutions to Total Variation Regularization Problems

In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.