Higher Degree Total Variation Research Articles

Motion-aligned 4D-MRI reconstruction using higher degree total variation and locally low-rank regularization

Four-dimensional magnetic resonance imaging (4D-MRI) is becoming increasingly important in radiotherapy treatment planning for its ability to simultaneously provide 3D structural information and temporal profiles of the examined tissues in a non-ionizing manner. However, the relatively long acquisition time and the resulting motion artifacts severely limit the further application of 4D-MRI. In this paper, we propose a novel motion-aligned reconstruction method based on higher degree total variation and locally low-rank regularization (maHDTV-LLR) to recover 4D MR images from the highly undersampled Fourier coefficients. Specifically, we propose a two-stage reconstruction framework alternating between a motion alignment step and a regularized optimization reconstruction step. Moreover, we incorporate the 3D-HDTV and the locally low-rank penalties into a unified framework to simultaneously exploit the spatial and temporal correlation of the 4D-MRI data. A fast alternating minimization algorithm based on variable splitting is utilized to solve the optimization problem efficiently. The performance of the proposed method is demonstrated in the context of 4D cardiac and abdominal MR images reconstruction with high undersampling factors. Numerical results show that the proposed method enables accelerated 4D-MRI with improved image quality and reduced artifacts.

An HDTV-SB imaging algorithm for wire-mesh tomography

In multiphase flow detection, wire-mesh sensors are an important detection tool. However, the reconstructed image is heavily jagged due to the rectangular measurement of the sensor structure. In order to overcome this shortcoming, an higher degree total variation (HDTV) algorithm based on a total variation imaging algorithm is applied to the image reconstruction of a wire-mesh sensor. This algorithm uses the directional derivative in each direction as a regularity and is applied to the equation with the smallest residual. Combining the HDTV algorithm with a split Bregman (SB) algorithm, the calculation speed of the algorithm is accelerated and, finally, a HDTV-SB image reconstruction algorithm which can be applied to the Jacobian matrix is obtained. Subsequently, we performed static simulation experiments of complex flow patterns and compared them with some traditional algorithms. Experiments show that the image reconstruction quality of the HDTV-SB algorithm is higher than other algorithms. Finally, quantitative experiments are carried out on the two important parameters of the HDTV-SB algorithm. The results show that if the parameters are adjusted within a certain range, a one-step to two-step iteration can obtain a better reconstructed image.

A mixed-order nonlinear diffusion compressed sensing MR image reconstruction.

Avoid formation of staircase artifacts in nonlinear diffusion-based MR image reconstruction without compromising computational speed. Whereas second-order diffusion encourages the evolution of pixel neighborhood with uniform intensities, fourth-order diffusion considers smooth region to be not necessarily a uniform intensity region but also a planar region. Therefore, a controlled application of fourth-order diffusivity function is used to encourage second-order diffusion to reconstruct the smooth regions of the image as a plane rather than a group of blocks, while not being strong enough to introduce the undesirable speckle effect. Proposed method is compared with second- and fourth-order nonlinear diffusion reconstruction, total variation (TV), total generalized variation, and higher degree TV using in vivo data sets for different undersampling levels with application to dictionary learning-based reconstruction. It is observed that the proposed technique preserves sharp boundaries in the image while preventing the formation of staircase artifacts in the regions of smoothly varying pixel intensities. It also shows reduced error measures compared with second-order nonlinear diffusion reconstruction or TV and converges faster than TV-based methods. Because nonlinear diffusion is known to be an effective alternative to TV for edge-preserving reconstruction, the crucial aspect of staircase artifact removal is addressed. Reconstruction is found to be stable for the experimentally determined range of fourth-order regularization parameter, and therefore not does not introduce a parameter search. Hence, the computational simplicity of second-order diffusion is retained.

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

We introduce a novel image regularization termed as multiple degree total variation (MDTV). This type of regularization combines the first and second degree directional derivatives, thus providing a good balance between preservation of edges and region smoothness. In order to solve the resulting optimization problem, we proposed a fast majorize minimize algorithm. We demonstrate the utility of the MDTV regularization in the context of image denoising and compressed sensing. We compare the proposed method with standard TV, and the state of the art higher degree methods, including higher degree total variation (HDTV) and total generalized variation (TGV) based schemes. Numerical results indicate that MDTV penalty provides improved image recovery performance.

A fast higher degree total variation minimization method for image restoration

Based on the spectral decomposition theory, this paper presents a unified analysis of higher degree total variation (HDTV) model for image restoration. Under this framework, HDTV is reinterpreted as a family of weighted L1–L2 mixed norms of image derivatives. Due to the equivalent formulation of HDTV, we construct a modified functional for HDTV-based image restoration. Then, the minimization of the modified functional can be decoupled into two separate sub-problems, which correspond to the deblurring and denoising. Thus, we design a fast and efficient image restoration algorithm using an iterative Wiener deconvolution with fast projected gradient denoising (IWD-FPGD) scheme. Moreover, we show the convergence of the proposed IWD-FPGD algorithm for the special case of second-degree total variation. Finally, the systematic performance comparisons of the proposed IWD-FPGD algorithm demonstrate the effectiveness in terms of peak signal-to-noise ratio, structural similarity and convergence rate.

Generalized Higher Degree Total Variation (HDTV) Regularization

We introduce a family of novel image regularization penalties called generalized higher degree total variation (HDTV). These penalties further extend our previously introduced HDTV penalties, which generalize the popular total variation (TV) penalty to incorporate higher degree image derivatives. We show that many of the proposed second degree extensions of TV are special cases or are closely approximated by a generalized HDTV penalty. Additionally, we propose a novel fast alternating minimization algorithm for solving image recovery problems with HDTV and generalized HDTV regularization. The new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper. Numerical experiments on 3D magnetic resonance images and 3D microscopy images show that HDTV and generalized HDTV improve the image quality significantly compared with TV.

A FAST MAJORIZE MINIMIZE ALGORITHM FOR HIGHER DEGREE TOTAL VARIATION REGULARIZATION.

The main focus of this paper is to introduce a computationally efficient algorithm for solving image recovery problems, regularized by the recently introduced higher degree total variation (HDTV) penalties. The anisotropic HDTV penalty is the fully separable L1 semi-norm of the directional image derivatives; the use of this penalty is seen to considerably improve image quality in biomedical inverse problems. We introduce a novel majorize minimize algorithm to solve the HDTV optimization problem, thus considerably speeding it over the previous implementation. Specifically, comparisons with previous iterative reweighted algorithm show an approximate ten fold speedup. The new algorithm enables us to obtain reconstructions that are free of patchy artifacts exhibited by classical TV schemes, while being comparable to state of the art total variation regularization schemes in run time.

Higher Degree Total Variation (HDTV) Regularization for Image Recovery

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.