You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We propose a new joint image reconstruction method by recovering edge directly from observed data. More specifically, we reformulate joint image reconstruction with vectorial total-variation regularization as an @math minimization problem of the Jacobian of the underlying multi-modality or multi-contrast images. Derivation of data fidelity for Jacobian and transformation of noise distribution are also detailed. The new minimization problem yields an optimal @math convergence rate, where @math is the iteration number, and the per-iteration cost is low thanks to the close-form matrix-valued shrinkage. We conducted numerical tests on a number multi-contrast magnetic resonance image (MRI) datasets, which show that the proposed method significantly improves reconstruction efficiency and accuracy compared to the state-of-the-arts.

Cited Abstract(s):
[1]: Vector-valued images such as RGB color images or multimodal medical images show a strong interchannel correlation, which is not exploited by most image processing tools. We propose a new notion of treating vector-valued images which is based on the angle between the spatial gradients of their channels. Through minimizing a cost functional that penalizes large angles, images with parallel level sets can be obtained. After formally introducing this idea and the corresponding cost functionals, we discuss their Gateaux derivatives that lead to a diffusion-like gradient descent scheme. We illustrate the properties of this cost functional by several examples in denoising and demosaicking of RGB color images. They show that parallel level sets are a suitable concept for color image enhancement. Demosaicking with parallel level sets gives visually perfect results for low noise levels. Furthermore, the proposed functional yields sharper images than the other approaches in comparison.

[2]: Combined positron emission tomography (PET) and magnetic resonance imaging (MRI) scanners acquire simultaneously functional PET and anatomical or functional MRI data. As the data of both modalities are likely to show similar structures we aim to exploit this by joint reconstruction of PET and MRI. In a Bayesian formulation, this can be achieved by adding prior information encoding that the images of the two modalities are not independent. Structural similarity can be modeled by the alignment of the image gradients or equivalently their level sets being parallel. Therefore we can combine the objective functions of both modalities and penalize image pairs which do not have parallel level sets. Our results show that combining the reconstruction from heavily under-sampled MRI and noisy PET data can lead to less under-sampling artifacts in MRI images and better defined PET images.

[3]: Inverse problems are inherently non-unique, and regularization is needed to obtain stable and reasonable solutions. The regularization adds information to the problem and determines which solution, out of the infinitely many, is obtained. In this paper, we review and discuss the case when a priori information exists in the form of either known structure or in the form of another inverse problem for a different property. The challenge is to include such information in the inversion process. To use existing known structure, we review the concept of model fusion, where we build a regularization functional that fuses the inverted model to a known one. The fusion is achieved by four different techniques. Joint inversion of two data sets is achieved by using iterative data fusion. The paper discusses four different methods for joint inversion. We discuss the use of correspondence maps or the petrophysics of the rocks, as well as structure. In particular, we suggest to further stabilize the well-known gradient cross product and suggest a new technique, Joint Total Variation, to solve the problem. The Joint Total Variation is a convex functional for joint inversion and, as such, has favorable optimization properties. We experiment with the techniques on the DC resistivity problem and the borehole tomography and show how model fusion and joint inversion can significantly improve over existing techniques.

[4]: Recent advances in technology have enabled the combination of positron emission tomography (PET) with magnetic resonance imaging (MRI). These PET-MRI scanners simultaneously acquire functional PET and anatomical or functional MRI data. As function and anatomy are not independent of one another the images to be reconstructed are likely to have shared structures. We aim to exploit this inherent structural similarity by reconstructing from both modalities in a joint reconstruction framework. The structural similarity between two modalities can be modelled in two different ways: edges are more likely to be at similar positions and or to have similar orientations. We analyse the diffusion process generated by minimizing priors that encapsulate these different models. It turns out that the class of parallel level set priors always corresponds to anisotropic diffusion which is sometimes forward and sometimes backward diffusion. We perform numerical experiments where we jointly reconstruct from blurred Radon data with Poisson noise (PET) and under-sampled Fourier data with Gaussian noise (MRI). Our results show that both modalities benefit from each other in areas of shared edge information. The joint reconstructions have less artefacts and sharper edges compared to separate reconstructions and the l2-error can be reduced in all of the considered cases of under-sampling.

[5]:
Besides joint TV or tensor based regularization, the parallelism of level sets across multi-modality contrast images are also proposed as joint image regularization in [1] [4] [5] [3] . The main idea is to exploit the structural similarity of two images @math and @math measured by the parallelism of their gradients @math and @math at each point using @math , for some functions @math and @math . Then the regularization in joint reconstruction takes form @math Several different choices of @math and @math have been studied in these works. For instance, in [2] , @math and @math are taken as identities, or @math and @math . In [3] , @math is the identity and @math . In [4] the side information on the level set, namely, the location and direction of the gradients of a reference image, is available to assist the reconstruction.