Time Frequency Analysis

The Fourier transform (FT) forms the cornerstone of magnetic resonance (MR) imaging.  However, the FT is not appropriate for signals whose frequency content change with time.  Since patient motion and physiological fluctuations can cause time-varying noise and artifacts, a time-frequency representation is more suitable MR signal processing.

Time domain signal containing two sine functions and two bursts of noise	Frequency domain of the signal generated from the Fourier Transform	Time-Frequency domain of the signal generated from the S-Transform

The Stockwell transform ( ST)  combines the time–frequency representation of the Gabor Transform with the multi-scaling feature of the Wavelet Transform. It provides a unique time–frequency representation of a signal by adapting the FT to analyze the localized signal, using frequency-dependent time-scaling windows. The interpretation in a time–frequency domain becomes much easier and the multi-resolution analysis allows the ST to detect subtle Fourier spectral changes over time. In addition, the Fourier and Stockwell spectra are intimately related; they can be readily converted from one to the other. This close connection suggests the possibility of pre-processing image data in the Stockwell domain for Fourier-based imaging modalities (in particular, MRI and CT). Thus, the properties of the ST make it a potentially valuable processing tool for medical imaging.

Projects at the centre related to time-frequency analysis include texture analysis in both multiple sclerosis and cancer patients, the development of new transforms, the formulation of an efficient (fast) ST and development of a general theoretical framework to describe time-frequency transforms and analysis.

Patents

• Local Multi-Scale Fourier Analysis for MRI. Mitchell JR, Fong C, Zhu H, Goodyear BG, Brown R. US patent #6,850,062 (issued Feb 1 2005).

Papers

• Texture Analysis for Non-Invasive Identification of Brain Tumor Genotype from MRI. Brown RA, Zlatescu MC, Cairncross JG, Mitchell JR. Proceedings of the Fifth IASTED International Conference on Visualization, Imaging, and Image Processing (VIIP). Benidorm, Spain. ACTA Press, #480-116 pg. 459-464 (2005).
  
• Distributed Vector Processing of a New Local Multi-Scale Fourier Transform For Medical Imaging Applications. Brown RA, Zhu H, Mitchell JR. IEEE Transactions on Medical Imaging, 24(5):689-91 (2005).
• 3D MRI Progressive Imaging: Data- and Transform-Space Strategies. Brown RA, Baeza I, Mitchell JR, Villanueva RJ, Zhu H, Villanueva-Oller J, Law AG. CCCT in-press.
• 3D Progressive Imaging with Feature Selection. Brown RA, Zhu H, Mitchell JR, Law AG. Proceedings of the 2004 International Conference on Mathematics and Engineering Techniques in Medicine and Biological Sciences (METMBS). Valafar FH, Arabnia HR, He M, Sinha U (Eds); CSREA Press, ISBN 1-932415-43-2, 216-219 (2004).
  
• A New Local Multiscale Fourier Analysis for Medical Imaging. Zhu H, Goodyear BG, Lauzon ML, Brown RA, Mayer G, Law AG, Mansinha L, Mitchell JR. Medical Physics, 30:1134-1141 (2003).

Invited Talks

• "Applications of time frequency analysis in medicine and geiophysics." Adler DH, Bjarnason TA, Drabycz S, Mitchell JR. ISAAC Workshop on Pseudo-Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Toronto, ON. December 11-15, 2006. (Podcast)
• Time/frequency analysis of medical images. Mitchell JR, Brown RA, Drabycz S. Pacific Institute for the Mathematical Sciences Workshop on Time-Frequency Analysis and Non-Stationary Filtering, Banff, Alberta, Canada (2005).
• Texture Analysis for Non-Invasive Identification of Brain Tumor Genotype from MRI. Brown RA, Zlatescu MC, Cairncross JG, Mitchell JR. Universidad Politécnica de Valencia, Valencia, Spain (2005).

Abstracts

• A 2D local frequency analysis approach using the S-transform. Drabycz S and Mitchell JR. Poster presentation by S Drabycz at the upcoming CAIMS/MITACS Joint Annual Conference, Toronto, Ontario (2006).

• Time/frequency analysis in magnetic resonance imaging. Drabycz S, Mitchell JR. Oral presentation at the Applied Mathematics Graduate Student Conference, Simon Fraser University, Vancouver, BC, Canada (2006).

• Progressive Imaging: A Transform Space Approach. Brown RA, Baeza I, Zhu H, Villanueva RJ, Mitchell JR, Law AG. Poster at the Society for Industrial and Applied Mathematics Conference, Salt Lake City, Utah (2004).
  

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