The adaptive filtering theory has been extensively developed, and most of the proposed algorithms work under the assumption of Euclidean space. However, in many applications, the data to be processed comes from a non-linear manifold. In this article, we propose an alternative adaptive filter that works on a manifold, thus generalizing the filtering task to non-Euclidean spaces. To this end, we generalized the least-mean-squared algorithm to work on a manifold using an exponential map. Our experiments showed that the proposed method outperforms other state-of-the-art algorithms in several filtering tasks.

The reconstruction of positron emission tomography data is a difficult task, particularly at low count rates because Poisson noise has a significant influence on the statistical uncertainty of positron emission tomography (PET) measurements. Prior information is frequently used to improve image quality. In this paper, we propose the use of a field of experts to model a priori structure and capture anatomical spatial dependencies of the PET images to address the problems of noise and low count data, which make the reconstruction of the image difficult. We reconstruct PET images by using a modified MXE algorithm, which minimizes a objective function with the cross-entropy as a fidelity term, while the field of expert model is incorporated as a regularizing term. Comparisons with the expectation maximization algorithm and a iterative method with a prior penalizing relative differences showed that the proposed method can lead to accurate estimation of the image, especially with acquisitions at low count rate.