Coimbra Institute for Biomedical Imaging and Translational Research

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Improving the conditioning of the method of fundamental solutions for the Helmholtz equation on domains in polar or elliptic coordinates

Publication . Antunes, Pedro R. S.; Calunga, Hernani; Serranho, Pedro

A new approach to overcome the ill-conditioning of the Method of Fundamental Solutions (MFS) combining Singular Value Decomposition (SVD) and an adequate change of basis was introduced in [1] as MFS-SVD. The original formulation considered polar coordinates and harmonic polynomials as basis functions and is restricted to the Laplace equation in 2D. In this work, we start by adapting the approach to the Helmholtz equation in 2D and later extending it to elliptic coordinates. As in the Laplace case, the approach in polar coordinates has very good numerical results both in terms of conditioning and accuracy for domains close to a disk but does not perform so well for other domains, such as an eccentric ellipse. We therefore consider the MFS-SVD approach in elliptic coordinates with Mathieu functions as basis functions for the latter. We illustrate the feasibility of the approach by numerical examples in both cases.

2024Journal article

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Texture for neuroimaging

Publication . Nunes, Ana; Serranho, Pedro; Castelo-Branco, Miguel; Bernardes, Rui

Texture analysis is an umbrella term for multiple image analysis techniques that quantify and characterize the distribution of the image’s gray levels. It has a natural application in biomedical image analysis, where texture-based techniques are increasingly being incorporated into neuroimaging research. In this chapter, the role of texture analysis in the field of neuroimaging is addressed. Neuroimaging applications of texture-based approaches are contextualized within past and recent developments in image texture analysis, the categories of texture analysis methods, and the typical texture-based problem types, namely, classification and segmentation. Neuroimaging applications using magnetic resonance imaging, positron emission tomography, and optical coherence tomography are individually reviewed, and some considerations on the future perspectives for texture-based approaches in neuroimaging are made.

2024-06-13Book part

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