Spherical deconvolution of multichannel diffusion MRI data with non-Gaussian noise models and spatial regularization

Spherical deconvolution (SD) methods are widely used to estimate the intra-voxel white-matter fiber orientations from diffusion MRI data. However, while some of these methods assume a zero-mean Gaussian distribution for the underlying noise, its real distribution is known to be non-Gaussian and to depend on the methodology used to combine multichannel signals. Indeed, the two prevailing methods for multichannel signal combination lead to Rician and noncentral Chi noise distributions. Here we develop a Robust and Unbiased Model-BAsed Spherical Deconvolution (RUMBA-SD) technique, intended to deal with realistic MRI noise, based on a Richardson-Lucy (RL) algorithm adapted to Rician and noncentral Chi likelihood models. To quantify the benefits of using proper noise models, RUMBA-SD was compared with dRL-SD, a well-established method based on the RL algorithm for Gaussian noise. Another aim of the study was to quantify the impact of including a total variation (TV) spatial regularization term in the estimation framework. To do this, we developed TV spatially-regularized versions of both RUMBA-SD and dRL-SD algorithms. The evaluation was performed by comparing various quality metrics on 132 three-dimensional synthetic phantoms involving different inter-fiber angles and volume fractions, which were contaminated with noise mimicking patterns generated by data processing in multichannel scanners. The results demonstrate that the inclusion of proper likelihood models leads to an increased ability to resolve fiber crossings with smaller inter-fiber angles and to better detect non-dominant fibers. The inclusion of TV regularization dramatically improved the resolution power of both techniques. The above findings were also verified in brain data

Enhancing Compressed Sensing 4D Photoacoustic Tomography by Simultaneous Motion Estimation

A crucial limitation of current high-resolution 3D photoacoustic tomography (PAT) devices that employ sequential scanning is their long acquisition time. In previous work, we demonstrated how to use compressed sensing techniques to improve upon this: images with good spatial resolution and contrast can be obtained from suitably sub-sampled PAT data acquired by novel acoustic scanning systems if sparsity-constrained image reconstruction techniques such as total variation regularization are used. Now, we show how a further increase of image quality can be achieved for imaging dynamic processes in living tissue (4D PAT). The key idea is to exploit the additional temporal redundancy of the data by coupling the previously used spatial image reconstruction models with sparsity-constrained motion estimation models. While simulated data from a two-dimensional numerical phantom will be used to illustrate the main properties of this recently developed joint-image-reconstruction-and-motion-estimation framework, measured data from a dynamic experimental phantom will also be used to demonstrate their potential for challenging, large-scale, real-world, three-dimensional scenarios. The latter only becomes feasible if a carefully designed combination of tailored optimization schemes is employed, which we describe and examine in more detail

A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure

Total Directional Variation for Video Denoising

In this paper, we propose a variational approach for video denoising, based on a total directional variation (TDV) regulariser proposed in Parisotto et al. (2018), for image denoising and interpolation. In the TDV regulariser, the underlying image structure is encoded by means of weighted derivatives so as to enhance the anisotropic structures in images, e.g. stripes or curves with a dominant local directionality. For the extension of TDV to video denoising, the space-time structure is captured by the volumetric structure tensor guiding the smoothing process. We discuss this and present our whole video denoising work-flow. Our numerical results are compared with some state-of-the-art video denoising methods.SP acknowledges UK EPSRC grant EP/L016516/1 for the CCA DTC. CBS acknowledges support from Leverhulme Trust project on Breaking the non-convexity barrier, EPSRC grant Nr. EP/M00483X/1, the EPSRC Centre EP/N014588/1, the RISE projects CHiPS and NoMADS, the CCIMI and the Alan Turing Institute

Review of the mathematical foundations of data fusion techniques in surface metrology

The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed

An overview of the fundamental approaches that yield several image denoising techniques

Digital image is considered as a powerful tool to carry and transmit information between people. Thus, it attracts the attention of large number of researchers, among them those interested in preserving the image features from any factors that may reduce the image quality. One of these factors is the noise which affects the visual aspect of the image and makes others image processing more difficult. Thus far, solving this noise problem remains a challenge for the researchers in this field. A lot of image denoising techniques have been introduced in order to remove the noise by taking care of the image features; in other words, getting the best similarity to the original image from the noisy one. However, the findings are still inconclusive. Beside the enormous amount of researches and studies which adopt several mathematical concepts (statistics, probabilities, modeling, PDEs, wavelet, fuzzy logic, etc.), there is also the scarcity of review papers which carry an important role in the development and progress of research. Thus, this review paper intorduce an overview of the different fundamental approaches that yield the several image-denoising techniques, presented with a new classification. Furthermore, the paper presents the different evaluation tools needed on the comparison between these techniques in order to facilitate the processing of this noise problem, among a great diversity of techniques and concepts