A generalized Condat's algorithm of 1D total variation regularization

To solve the ill-posed problem in electrical impedance tomography (EIT) and overcome the blur effect made by conventional L2 Tikhonov regularization, a new two-step iterative regularization algorithm is applied for the process reconstruction of lung respiration. This algorithm is based on total variation (TV) regularization method, and uses a two-step iterative method with TV denoising operator, so that it can achieve double regularization effect. This algorithm has advantages in both resolution and robustness. Both simulation and experiment results are executed to prove the effectiveness of TV regularization method.

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.

Efficient general sparse denoising with non-convex sparse constraint and total variation regularization

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.

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.

Magnetic resonance imaging (MRI) reconstruction model based on total variation (TV) regularization can deal with problems such as incomplete reconstruction, blurred boundary, and residual noise. In this article, a non‐convex isotropic TV regularization reconstruction model is proposed to overcome the drawback. Moreau envelope and minmax‐concave penalty are firstly used to construct the non‐convex regularization of L2 norm, then it is applied into the TV regularization to construct the sparse reconstruction model. The proposed model can extract the edge contour of the target effectively since it can avoid the underestimation of larger nonzero elements in convex regularization. In addition, the global convexity of the cost function can be guaranteed under certain conditions. Then, an efficient algorithm such as alternating direction method of multipliers is proposed to solve the new cost function. Experimental results show that, compared with several typical image reconstruction methods, the proposed model performs better. Both the relative error and the peak signal‐to‐noise ratio are significantly improved, and the reconstructed images also show better visual effects. The competitive experimental results indicate that the proposed approach is not limited to MRI reconstruction, but it is general enough to be used in other fields with natural images.

Spectral unmixing aims at identifying the pure spectral signatures in hyperspectral images and simultaneously estimating their proportions in each pixel of the scene. By using an available spectral library as a dictionary, sparse-regression-based approaches aim at finding a subset of the dictionary that can optimally model each pixel in a given hyperspectral image. regularizer has been widely considered as a regularization strategy to exploit the sparsity of the unmixing solution. Further sparsity can be imposed by also using weighting factors. However, most existing strategies focus on the unmixing solution ignoring the gradient information. To account for the gradient information in hyperspectral unmixing, we propose a weighted sparse regression with total variation (WSRTV) unmixing model. The proposed WSRTV model incorporates gradient information in the sparse regression formulation by means of the weighted total variation (WTV) regularizer. The model imposes sparsity on both the solution and the gradient to improve the performance of unmixing. A dual symmetric Gauss-Seidel alternating direction method of multipliers (sGSADMM) is designed to optimize the proposed model. The designed algorithm both handles the anisotropic and isotropic WTV. Simulated and real hyperspectral data demonstrate the effectiveness of the proposed framework.

Nuclear magnetic resonance imaging (MRI) has become the gold standard for many imaging applications. It is routinely used to detect diseases and tissue pathologies due to its superior soft tissue contrast and high spatial resolution. Unlike other imaging techniques such as X-rays, computed tomography, it does not harm patients with ionising radiation.However, MRIs are expensive and only a small portion of patients who would benefit from them can access them. The pursuit of higher sensitivity has generated a trend towards increasing the already large static magnetic field. This has improved imaging quality but has made this technology more expensive and further hindered MRI accessibility. The main cost driver of MRI machines is the strong magnet.The collection of techniques being developed to overcome the intrinsic low NMR sensitivity has fostered interest in low field (LF) MRI to improve cost-effectiveness of MRI. First, the purchase and operational cost of LF systems is relatively low. Second, they are more compatible with other instrumentation which enables versatile concepts such as application-specific open systems and intraoperative imaging. LF systems can offer a wide range of configurations by using permanent magnets. Third, the small fringe field lowers the risk to patients and workers and makes the magnet easier to site. Fourth, artefacts which rely on the strength of field are reduced, as well as the specific absorption rate (SAR) in tissues.Ultra-low field (ULF) MRI systems employ field strengths lower than 10 mT. These systems can complement the advantages of LF systems mentioned above with additional potential benefits. The requirement of homogeneity in ULF is relaxed extensively. It is possible to adjust the strength and direction of the measurement field, allowing flexibility in pulse sequences. Moreover, ULF systems operate in the low kHz frequency range, avoiding the need for radio frequency (RF) noise shielding and enhancing T1-weighted contrast.Most ULF MRI research concentrates on using expensive and fragile sensors, produces the magnetic fields through resistive coils and employs linear gradient fields for encoding. Although ULF systems are important low-cost and portable MRI candidates, they still need considerable space to generate linear fields and need expensive and bulky high permeability shielding to suppress ambient noise.This PhD project develops two major solutions towards a portable and low-cost ULF MRI system: a software gradiometer based on machine learning for ambient noise cancellation and a fast iterative algorithm with total variation (TV) regularisation for image reconstruction. In addition, a concept for a dynamical permanent magnet array (PMA) ULF MRI system is proposed with nonlinear 3D spatial encoding.First, I introduce an experimental setup as a platform for evaluating the MR signal in the ULF regime using a Helmholtz coil as a framework. It is composed of a measurement field, an NMR console, a transmit and receive coil, and a shielding box. The NMR signal is successfully acquired and analysed in partial shielding and full shielding. The results suggest that partial shielding of ULF MRI instruments has the potential to generate a strong signal. This system is built for a subsequent study on noise cancellation presented in following chapter.The second study is a convolutional neural network (CNN) based software gradiometer for background noise cancellation. It includes three ambient noise monitoring coils which are placed away from the NMR sensing coil. The noise induced in the sensing coils is estimated using machine learning. The results show that the software gradiometer reduces ambient noise and can be used to relax shielding requirements, thereby reducing the size and weight of the shielding box.The concept of using small magnets for 3D-spatial encoding in dynamical PMA based ULF MRI is described. It uses small rectangular permanent magnets to generate a 3D-spatial encoding field. The encoding magnets move along an optimised trajectory to achieve an encoding matrix with a low condition number. A prototype of the 3D-spatial encoding has been built and its feasibility tested. The described encoding system can reduce the volume and energy requirements and improve the portability of ULF MRI systems.Finally, iterative imaging reconstruction algorithms for ULF MR are studied. The current state-of-the-art methods are evaluated, which includes Simultaneous Iterative Reconstruction Technique (ART/SIRT), Conjugate Gradient Least Squares (CGLS), and the Chambolle-Pock (CP) algorithms. A dynamic weight Column Action Reconstruction Technique (dwCART) method is proposed and applied to image reconstruction with TV regularisation in the setting of undersampling. In the absence of having a finalised prototype to acquire real ULF MRI imaging data, the methods are tested on CT models (using both simulated and real CT data). The dwCART algorithm performed best on the tested CT problems. It paves the way for fast and accurate imaging with the PMA-based ULF MRI system.

The total variation (TV) regularization method has been used to solve the ill-posed inverse problem of electrical resistance tomography (ERT), owing to its good ability to preserve edges. However, the quality of the reconstructed images, especially in the flat region, is often degraded by noise. To optimize the regularization term and the regularization factor according to the spatial feature and to improve the resolution of reconstructed images, a spatially adaptive total variation (SATV) regularization method is proposed. A kind of effective spatial feature indicator named difference curvature is used to identify which region is a flat or edge region. According to different spatial features, the SATV regularization method can automatically adjust both the regularization term and regularization factor. At edge regions, the regularization term is approximate to the TV functional to preserve the edges; in flat regions, it is approximate to the first-order Tikhonov (FOT) functional to make the solution stable. Meanwhile, the adaptive regularization factor determined by the spatial feature is used to constrain the regularization strength of the SATV regularization method for different regions. Besides, a numerical scheme is adopted for the implementation of the second derivatives of difference curvature to improve the numerical stability. Several reconstruction image metrics are used to quantitatively evaluate the performance of the reconstructed results. Both simulation and experimental results indicate that, compared with the TV (mean relative error 0.288, mean correlation coefficient 0.627) and FOT (mean relative error 0.295, mean correlation coefficient 0.638) regularization methods, the proposed SATV (mean relative error 0.259, mean correlation coefficient 0.738) regularization method can endure a relatively high level of noise and improve the resolution of reconstructed images.

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.

The use of non-local self-similarity prior between image blocks can improve image reconstruction performance significantly. We propose a compressive sensing image reconstruction algorithm that combines bilateral total variation and nonlocal low-rank regularization to overcome over-smoothing and degradation of edge information which result from the prior reconstructed image. The proposed algorithm makes use of the preservation of image edge information by bilateral total variation operator to enhance the edge details of the reconstructed image. In addition, we use weighted nuclear norm regularization as a low-rank constraint for similar blocks of the image. To solve this convex optimization problem, the Alternating Direction Method of Multipliers (ADMM) is employed to optimize and iterate the algorithm model effectively. Experimental results show that the proposed algorithm can obtain better image reconstruction quality than conventional algorithms with using total variation regularization or considering the nonlocal structure of the image only. At 10% sampling rate, the peak signal-to-noise ratio gain is up to 2.39 dB in noiseless measurements compared with Nonlocal Low-rank Regularization (NLR-CS). Reconstructed image comparison shows that the proposed algorithm retains more high frequency components. In noisy measurements, the proposed algorithm is robust to noise and the reconstructed image retains more detail information.

In this paper, we propose a variational model for Polarimetric synthetic aperture radar (PolSAR) data speckle reduction. Since the statistical characteristic of covariance matrix can be well described by the complex Wishart distribution, a data fitting term can be derived according to the maximum likelihood rule. Combining it with the total variation (TV) regularization gives a variational model for PolSAR image speckle reduction. An algorithm based on the alternating minimization technique is used to solve the variational problem. Experimental results on real PolSAR images demonstrate the validity of the proposed method.

In the field of magnetic anomaly detection (MAD), the anomaly signal is easily submerged by ambient electromagnetic interference. Though the existing noise suppression methods can effectively improve the signal-to-noise ratio (SNR), there are still some intractable problems, such as signal distortion and boundary blur. To solve these problems, a novel MAD method based on structured low rank (SLR) and total variation (TV) regularization constraints is proposed in this letter. The noise suppression performance is improved by leveraging the structured low rankness of the signal. To preserve clean boundaries of the anomalies, an anisotropic TV regularization constraint is employed in the approach. Comparing the SLR-TV method with four state-of-the-art methods with extensive field tests, the results demonstrate that the proposed SLR-TV method achieves the greatest SNR improvement by about 63.24% and the best structural similarity (SSIM) improvement by about 53.02% over other methods in the range from −40 to 0 dB, showing the utility and high fidelity of the proposed framework in low SNR.

Image denoising by means of total variation (TV) regularization is still a standard procedure. For very large images, especially three-dimensional voxel datasets, however, this can be computationally infeasible. We show how this TV regularization can be approximately performed even in arbitrary dimensions by applying appropriate shrinkage to selected and properly weighted Haar wavelet coefficients, all of which depends even on the dimensionality of the data. Our approach acts entirely on the wavelet coefficients which represent the compressed image, and is therefore suited for the application on large three-dimensional images represented in the Haar wavelet basis, e.g., volumes from computed tomography.

Application of Regularized Richardson-Lucy Algorithm for Deconvolution of Confocal Microscopy Images