# Domain-Informed Signal Processing with Application to Analysis of Human Brain Functional MRI Data

Sammanfattning: Standard signal processing techniques are implicitly based on the assumption that the signal lies on a regular, homogeneous domain. In practice, however, many signals lie on an irregular or inhomogeneous domain. An application area where data are naturally defined on an irregular or inhomogeneous domain is human brain neuroimaging. The goal in neuroimaging is to map the structure and function of the brain using imaging techniques. In particular, functional magnetic resonance imaging (fMRI) is a technique that is conventionally used in non-invasive probing of human brain function. This doctoral dissertation deals with the development of signal processing schemes that adapt to the domain of the signal. It consists of four papers that in different ways deal with exploiting knowledge of the signal domain to enhance the processing of signals. In each paper, special focus is given to the analysis of brain fMRI data, either as the main theme (Paper I) or as proof of practical significance of the proposed schemes (Papers II, III and IV). Paper I presents a framework for enhanced fMRI activation mapping through exploiting filters that adapt to the brain anatomy. A novel procedure for constructing brain graphs, with subgraphs that separately encode the topology of the cerebral and cerebellar gray matter, is presented. Graph wavelets tailored to the convoluted boundaries of brain gray matter are designed and exploited to implement an anatomically-informed spatial transformation on fMRI data. Compared to conventional brain activation mapping schemes, the proposed approach shows superior type-I error control. Results on real data suggest a higher detection sensitivity as well as capability to capture subtle, connected patterns of brain activity. Paper II presents a graph-based signal decomposition scheme that adapts to the domain of the data as well as to the spectral content of a given signal set. The construction starts from the design of a prototype Meyer-type system of kernels with uniform subbands. The adaptivity of the approach is introduced by exploiting the ensemble energy spectral density. Using the ensemble energy spectral density, the prototype design is warped such that the resulting subbands each capture an equal amount of energy for the given signal class. Results on fMRI data and Monte Carlo simulations illustrate the superiority of signal-adapted frames over frames blind to signal characteristics in representing data and in denoising. Paper III presents a generic interpolation scheme for reconstructing signal samples from an inhomogeneous domain. The interpolation adapts to the inhomogeneity of the domain. The adaptation is incorporated by introducing a domain-similarity metric that characterises the domain in the adjacency of each sample point. The interpolation is shown to satisfy the domain-informed consistency principle, a principle that we define as an extension of the classical consistency principle. As proof of concept, domain-informed linear interpolation is presented as an extension of standard linear interpolation. Results from applying the proposed approach on fMRI data demonstrated its potential to reveal subtle details. Paper IV extends the theory in Paper III to enable reconstruction of signals with varying degrees of spatial smoothness. In particular, conventional shift-invariant B-spline interpolation is extended to a shift-variant, domain-informed interpolation. This is done by constructing a domain-informed generating basis that satisfies stability properties. The benefit of domain-informed interpolation over standard B-spline interpolation is demonstrated through Monte Carlo simulations across a range of B-spline orders. The practical significance of domain-informed spline interpolation is demonstrated on fMRI data. The results show the benefit of incorporating domain knowledge so that an interpolant consistent to the anatomy of the brain can be recovered by the proposed interpolation.

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