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Harmonic Phase Processing In Magnetic Resonance Susceptibility Imaging Open Access


Other title
phase imaging
harmonic extension
susceptibility mapping
magnetic resonance imaging
Type of item
Degree grantor
University of Alberta
Author or creator
Topfer, Ryan JS
Supervisor and department
Wilman, Alan (Biomedical Engineering)
Examining committee member and department
Sacchi, Mauricio (Physics)
De Zanche, Nicola (Oncology)
Wilman, Alan (Biomedical Engineering)
Department of Biomedical Engineering

Date accepted
Graduation date
Master of Science
Degree level
The popularity of magnetic resonance imaging (MRI) owes much to its flexibility. Sensitive to a host of different biophysical phenomena, parameters of the scan can be fine-tuned to highlight specific pathologies. One such mechanism for generating image contrast is magnetic susceptibility—the material property that defines how an object will distort an applied magnetic field such as that of the MR scanner. In functional MRI (fMRI), for instance, the unique magnetic signatures of oxygenated (diamagnetic) and deoxygenated (paramagnetic) blood are what permit the indirect measure of neuronal activity. However, rather than measuring the susceptibility itself, fMRI registers haemodynamic change as subtle gains and losses in the signal magnitude due to haem-iron induced dephasing of the proton spins. Susceptibility mapping, an emerging area in MRI, attempts to retrieve a direct measure of bulk susceptibility itself. Generally this is done, not by means of the standard magnitude image, but through the phase component of the signal which, in the idealized case, relates the magnetic field perturbation by a simple multiplicative constant. Several issues interfere with the construction of accurate susceptibility maps. Foremost is the obfuscation of the small-scale “local” field (i.e., that pertaining to susceptibility vari- ation within tissue) by the so-called “background” field, which owes predominantly to the comparatively substantial susceptibility shift between tissue and air. Whether the goal is to produce qualitatively useful images of the local field, or to map the susceptibility itself, the local field must first be isolated from the background. To isolate overlapping signals which are a priori unknown, the unique characteristics of the expected signals need to first be cod- ified. One family of methods for isolating the local field begins by asserting that, away from air-tissue interfaces, the background field should satisfy the partial differential equation of Laplace, whereas the local field should satisfy that of Poisson (viz., the background field should be harmonic, the local field non-harmonic). This classification informs a filtering technique based on the spherical mean value (SMV) property of harmonic functions: the mean of a harmonic function calculated over a spherical surface equates to the specific value taken by the function at the centre of the sphere. This thesis introduces another property of harmonic functions to the task of local field estimation: a harmonic function can be locally expressed by means of a convergent power series (viz., it is an analytic function). This property is first employed to characterize the SMV kernel as an estimator for the central field value when the field is discretized. Analysis reveals that when the field data is of a finite spatial resolution, the aim of accurate elimination of the background field via the SMV is fundamentally at odds with the aim of preserving the local field. Fortunately, given the rapid decay of the background field and its derivatives with distance from its field sources, the discrete SMV is nevertheless a robust estimator for field geometries such as those observed in the brain. In addition to the problem of finite image resolution, MR imaging of the head has finite spatial support as signal is generally absent in the skull and ever-absent in the surrounding air. The SMV cannot be used to estimate the background field wherever the spherical kernel extends beyond the edges of this support. Hence, in conventional SMV-filtering, field points within a distance from the edges equal to the radius of the employed kernel are discarded outright. This is a considerable obstacle to a number of clinical applications as it precludes analysis of features such as subdural haematomas and cortical lesions in pathologies such as multiple sclerosis. To recover local field across the edges of the brain, this study presents an extension to conventional SMV-filtering by appealing to the analytic nature of the back- ground field: by obtaining an initial SMV-estimate of the harmonic background field over a reduced inner portion of the brain, a three-dimensional Taylor expansion was performed to extend field coverage to the edges of the brain. The method is quantitatively assessed through a numerical experiment and qualitatively demonstrated on in vivo human brain data acquired at 4.7 T. Using a kernel radius typical of conventional methods, the extension recovered on average 26 % more in vivo brain volume.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
Topfer R, Schweser F, Deistung A, Reichenbach JR, Wilman AH. SHARP Edges: Recovering Cortical Phase Contrast Through Harmonic Extension. Magnetic Resonance in Medicine; article first published online March 3, 2014 DOI:10.1002/mrm.25148.

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