| 1. |
- Bergman Ärlebäck, Jonas, 1972-, et al.
(författare)
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The Bach tensor and other divergence-free tensors
- 2005
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Ingår i: International Journal of Geometric Methods in Modern Physics (IJGMMP). - 0219-8878. ; 2:1, s. 13-21
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Tidskriftsartikel (refereegranskat)abstract
- In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.
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| 2. |
- Boito, Deneb, 1993-
(författare)
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Diffusion MRI with generalised gradient waveforms : methods, models, and neuroimaging applications
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The incessant, random motion of water molecules within biological tissues reveals unique information about the tissues’ structural and functional characteristics. Diffusion magnetic resonance imaging is sensitive to this random motion, and since the mid-1990s it has been extensively employed for studying the human brain. Most notably, measurements of water diffusion allow for the early detection of ischaemic stroke and for the unveiling of the brain’s wiring via reconstruction of the neuronal connections. Ultimately, the goal is to employ this imaging technique to perform non-invasive, in vivo virtual histology to directly characterise both healthy and diseased tissue. Recent developments in the field have introduced new ways to measure the diffusion process in clinically feasible settings. These new measurements, performed by employing generalised magnetic field gradient waveforms, grant access to specific features of the cellular composition and structural organisation of the tissue. Methods based on them have already proven beneficial for the assessment of different brain diseases, sparking interest in translating such techniques into clinical practice. This thesis focuses on improving the methods currently employed for the analysis of such diffusion MRI data, with the aim of facilitating their clinical adoption. The first two publications introduce constrained frameworks for the estimation of parameters from diffusion MRI data acquired with generalised gradient waveforms. The constraints are dictated by mathematical and physical properties of a multi-compartment model used to represent the brain tissue, and can be efficiently enforced by employing a relatively new optimisation scheme called semidefinite programming. The developed routines are demonstrated to improve robustness to noise and imperfect data collection. Moreover, constraining the fit is shown to relax the requirements on the number of points needed for the estimation, thus allowing for faster data acquisition. In the third paper, the developed frameworks are employed to study the brain’s white matter in patients previously hospitalised for COVID-19 and who still suffer from neurological symptoms months after discharge. The results show widespread alterations to the structural integrity of their brain, with the metrics available through the advanced diffusion measurements providing new insights into the damage to the white matter. The fourth paper revisits the modelling paradigm currently adopted for the analysis of diffusion MRI data acquired with generalised gradient waveforms. Hitherto, the assumption of free diffusion has been employed to represent each domain in a multi-compartmental picture of the brain tissue. In this work, a model for restricted diffusion is considered instead to alleviate the paradoxical assumption of free but compartmentalised diffusion. The model is shown to perfectly capture restricted diffusion as measured with the generalised diffusion gradient waveforms, thus endorsing its use for representing each domain in the multi-compartmental model of the tissue.
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| 3. |
- Boito, Deneb, 1993-, et al.
(författare)
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Diffusivity-limited q-space trajectory imaging
- 2023
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Ingår i: Magnetic Resonance Letters. - : KeAi Publishing Communications. - 2772-5162. ; 3:2, s. 187-196
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Tidskriftsartikel (refereegranskat)abstract
- Q-space trajectory imaging (QTI) allows non-invasive estimation of microstructural features of heterogeneous porous media via diffusion magnetic resonance imaging performed with generalised gradient waveforms. A recently proposed constrained estimation framework, called QTI+, improved QTI’s resilience to noise and data sparsity, thus increasing the reliability of the method by enforcing relevant positivity constraints. In this work we consider expanding the set of constraints to be applied during the fitting of the QTI model. We show that the additional conditions, which introduce an upper bound on the diffusivity values, further improve the retrieved parameters on a publicly available human brain dataset as well as on data acquired from healthy volunteers using a scanner-ready protocol.
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| 4. |
- Brun, Anders, 1976-, et al.
(författare)
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Fast manifold learning based on Riemannian normal coordinates
- 2005
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Ingår i: Image Analysis. - Berlin, Heidelberg : Springer Berlin/Heidelberg. - 9783540263203 - 9783540315667 ; , s. 920-
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Konferensbidrag (refereegranskat)abstract
- We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra’s algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.
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| 5. |
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| 6. |
- Brun, Anders, 1976-, et al.
(författare)
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Intrinsic and Extrinsic Means on the Circle -- a Maximum Likelihood Interpretation
- 2007
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Ingår i: ICASSP 2007. IEEE International Conference on Acoustics, Speech and Signal Processing, 2007. - New York, USA : IEEE. - 1424407273 ; , s. III-1053-III-1056
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Konferensbidrag (refereegranskat)abstract
- For data samples in Rn, the mean is a well known estimator. When the data set belongs to an embedded manifold M in Rn, e.g. the unit circle in R2, the definition of a mean can be extended and constrained to M by choosing either the intrinsic Riemannian metric of the manifold or the extrinsic metric of the embedding space. A common view has been that extrinsic means are approximate solutions to the intrinsic mean problem. This paper study both means on the unit circle and reveal how they are related to the ML estimate of independent samples generated from a Brownian distribution. The conclusion is that on the circle, intrinsic and extrinsic means are maximum likelihood estimators in the limits of high SNR and low SNR respectively
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| 8. |
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| 9. |
- Brun, Anders, 1976-, et al.
(författare)
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Using Importance Sampling for Bayesian Feature Space Filtering
- 2007
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Ingår i: Proceedings of the 15th Scandinavian conference on image analysis. - Berlin, Heidelberg : Springer-Verlag. - 9783540730392 ; , s. 818-827
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Konferensbidrag (refereegranskat)abstract
- We present a one-pass framework for filtering vector-valued images and unordered sets of data points in an N-dimensional feature space. It is based on a local Bayesian framework, previously developed for scalar images, where estimates are computed using expectation values and histograms. In this paper we extended this framework to handle N-dimensional data. To avoid the curse of dimensionality, it uses importance sampling instead of histograms to represent probability density functions. In this novel computational framework we are able to efficiently filter both vector-valued images and data, similar to e.g. the well-known bilateral, median and mean shift filters.
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