# Projects

## Brain fMRI activation mapping using graph wavelets

This project explores a novel wavelet-based methodology for the purpose of analyzing brain functional MRI (fMRI) data with the help of graph theory. We define graphs based on the structural connectivity of the brain and construct wavelets that adapt to the convoluted structure of the brain’s grey matter; this is with the aim to gain a higher sensitivity and spatial accuracy in detecting neural activity that is known to lie within grey matter. The project is in collaboration with the Medical Image Processing Lab at EPFL, Geneva, Switzerland.

For more details, please refer to the following article:

H. Behjat, N. Leonardi, L. Sörnmo, D. Van De Ville, “Anatomically-adapted graph wavelets for improved group-level fMRI activation mapping,” NeuroImage, vol. 123, pp. 185-199, 2015.

## Design of signal-adapted tight frames on graphs

Analysis of signals on complex topologies modeled by graphs is a topic of increasing importance. Decompositions play a crucial role in representation and processing of such information. We propose a new tight frame design that is adapted to the spectral characteristics of a given class of signals defined on a given graph. The idea is to construct an ''adapted'' frame, in the sense of taking into account the energy-wise significance of the eigenvalues, rather than only adapting to the graph topology that is encoded by the eigenvalues as previously proposed. In this way, also the topological information of the graph is implicitly incorporated in the design, since the energy content is given in the graph spectral domain that is in turn defined by the eigenvalues. The proposed approach can be used to design tight frames for any given graph and signal set. Figure below shows an example construction for the Minnesota road graph. Using the designed frames in the context of denoising, the results (not shown here) suggest the superiority of the designed signal-adapted frames over frames blind to signal characteristics.

For more details, please refer to the following articles:

H. Behjat, U. Richter, D. Van De Ville and L. Sörnmo, “Signal-Adapted Tight Frames on Graphs,” IEEE Trans. Signal Process., vol. 64, no. 22, pp. 6017-6029, 2016 - can be downloaded from either here or here.

H. Behjat, D. Van De Ville, “Spectral design of signal-adapted tight frames on graphs,” In Vertex-Frequency Analysis of Graph Signals, ch. 4, pp. 177-206, Springer, 2019 - can be downloaded from either here or here.

## Interpolation in the presence of domain inhomogeneity

Standard interpolation techniques are implicitly based on the assumption that the signal lies on a homogeneous domain. We propose an interpolation scheme which exploits prior information about domain inhomogeneity, characterized by different, potentially overlapping, subdomains. In the 2-D setting, we denote this interpolation as domain-informed bilinear interpolation (DIBLI). The figure illustrates the setting for applying DIBLI on a 2-D slice of an fMRI volume, see (a), that accompanies a 3-fold higher resolution structural MRI scan, see (b). A close-up of an ROI marked in (b) is shown in (c). Segmenting the structural scan, one obtains GM, WM, and CSF probability maps, see (d)-(f); these maps are treated as normalized subdomain functions that fully describe the domain of the fMRI samples. A sample of column and row domain data for the marked position in (c), is illustrated in (h) and (i), respectively. A close-up of an ROI marked in (a) is shown in (g). Standard bilinear interpolation (SBLI) of the functional pixels shown in (g) is shown in (j). Two versions of DIBLI are shown in (k) and (l), the former with maximal and the later with minimal adaptation to domain knowledge. SBLI and both DIBLI versions are identical at homogeneous parts of the domain (see black arrows), whereas at the inhomogeneous parts (see white arrows), both DIBLI versions present finer details. Sharp signal boundaries at the intersection of subdomains can be recovered by DIBLI; the accuracy is limited by the level of discrepancy observed in the convoluted domain description. DIBLI with maximal adaptation provides further details over the minimal adapted version at some parts (see red arrows). On the other hand, DIBLI with minimal adaptation, cf. (l), may seem more visually appealing than (k), and yet, it presents significant subtle details that are missing in SBLI.

For further details and results, see:

H. Behjat, Z. Dogan, D. Van De Ville, L. Sörnmo, ''Domain-informed spline interpolation'', IEEE Trans. Signal Process., accepted with minor required revisions, revision submitted 4 Sep 2018 | arXiv:1810.07502.

## Localized Cortical Morphology Encoding Graphs

The human brain cortical layer has a convoluted morphology that is unique to each individual. Characterization of the cortical morphology is necessary in longitudinal studies of structural brain change, as well as in discriminating individuals in health and disease. A method for encoding the cortical morphology in the form of a graph is presented. The design of graphs that encode the global cerebral hemisphere cortices as well as localized cortical regions is proposed. Spectral metrics derived from these graphs are then studied and proposed as descriptors of cortical morphology. As proof-of-concept of their applicability in characterizing cortical morphology, the metrics are studied in the context of hemispheric asymmetry as well as gender dependent discrimination of cortical morphology.

For further details and results, see:

S. Maghsadhagh, A. Eklund and H. Behjat, “Graph Spectral Characterization of Brain Cortical Morphology,” arXiv:1902.07283.