The Fourier Transform in Biomedical Engineering / Edition 1

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This volume is designed to de-mystify the concept of Fourier transform s and to present practical and relevant applications of this tool and some of its relatives, such as wavelets, in biomedical situations - no tably in medical imaging. One-, two- and multidimensional Fourier tran sforms and wavelets are examined using intuitive, real-world examples in addition to a comprehensive review of their rigorous mathematical f oundation. Magnetic Resonance Imaging (MRI) is thoroughly explained, a s well as computed tomography. The book is an ideal text for junior en gineering and senior life science students. It will also be an excelle nt self-study source for biomedical technologists, physicians and othe r healthcare professionals.

Incl. use in MRI; wavelet transform; electrophysiological signal analysis; discrete & continuous Fourier transform etc

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Product Details

  • ISBN-13: 9780817639419
  • Publisher: Birkhauser Verlag
  • Publication date: 3/28/1998
  • Series: Applied and Numerical Harmonic Analysis Series
  • Edition description: 1998
  • Edition number: 1
  • Pages: 199
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.56 (d)

Table of Contents

1 Introduction to the Fourier Transform.- 1.1 Introduction.- 1.2 Basic Functions.- 1.3 Sines, Cosines and Composite waves.- 1.4 Orthogonality.- 1.5 Waves in time and space.- 1.6 Complex numbers. A Mathematical Tool.- 1.7 The Fourier transform.- 1.8 Fourier transforms in the physical world: The Lens as an FT computer.- 1.9 Blurring and convolution.- 1.9.1 Blurring.- 1.9.2 Convolution.- 1.10 The “Point” or “Impulse” response function..- 1.11 Band-limited functions.- 1.12 Summary.- 1.13 Bibliography.- 2 The 1-D Fourier Transform.- 2.1 Introduction.- 2.2 Re-visiting the Fourier transform.- 2.3 The Sampling Theorem.- 2.4 Aliasing.- 2.5 Convolution.- 2.6 Digital Filtering.- 2.7 The Power Spectrum.- 2.8 Deconvolution.- 2.9 System Identification.- 2.10 Summary.- 2.11 Bibliography.- 3 The 2-D Fourier Transform.- 3.1 Introduction.- 3.2 Linear space-invariant systems in two dimensions.- 3.3 Ideal systems.- 3.4 A simple X-ray imaging system.- 3.5 Modulation Transfer Function (MTF).- 3.6 Image processing.- 3.7 Tomography.- 3.8 Computed Tomography.- 3.9 Summary.- 3.10 Bibliography.- 4 The Fourier Transform in Magnetic Resonance Imaging.- 4.1 Introduction.- 4.2 The 2-D Fourier transform.- 4.3 Magnetic Resonance Imaging.- 4.3.1 Nuclear Magnetic Resonance.- 4.3.2 Excitation, Evolution, and Detection.- 4.3.3 The Received Signal: FIDs and Echos.- 4.4 MRI.- 4.4.1 Localization: Magnetic Field Gradients.- 4.4.2 The MRI Signal Equation.- 4.4.3 2-D Spin-Warp Imaging.- 4.4.4 Fourier Sampling: Resolution, Field-of-View, and Aliasing.- 4.4.5 2-D Multi-slice and 3-D Spin Warp Imaging.- 4.4.6 Alternate k -space Sampling Strategies.- 4.5 Magnetic Resonance Spectroscopic Imaging.- 4.5.1 Nuclear Magnetic Resonance Spectroscopy: 1-D.- 4.5.2 Magnetic Resonance Spectroscopic Imaging: 2-D, 3-D, and 4-D.- 4.6 Motion in MRI.- 4.6.1 Phase Contrast Velocity Imaging.- 4.6.2 Phase Contrast Angiography.- 4.7 Conclusion.- 4.8 Bibliography.- 5 The Wavelet Transform.- 5.1 Introduction.- 5.1.1 Frequency analysis: Fourier transform.- 5.2 Time-Frequency analysis.- 5.2.1 Generalities.- 5.2.2. How does time-frequency analysis work?.- 5.2.3 Windowed Fourier transform.- 5.2.4 Wavelet transform.- 5.3 Multiresolution Analysis.- 5.3.1 Scaling Functions.- 5.3.2 Definition.- 5.3.3 Scaling Relation.- 5.3.4 Relationship of multiresolution analysis to wavelets.- 5.3.5 Multiresolution signal decomposition.- 5.3.6 Digital filter interpretation.- 5.3.7 Fast Wavelet Transform Algorithm.- 5.3.8 Multidimensional Wavelet Transforms.- 5.3.9 Fourier vs. Wavelet Digital Signal Processing.- 5.4 Applications.- 5.4.1 Image Compression.- 5.4.2 Irregular heart beat detection from EKG signals.- 5.5 Summary.- 5.6 Bibliography.- 6 The Discrete Fourier Transform and Fast Fourier Transform.- 6.1 Introduction.- 6.2 From Continuous to Discrete.- 6.2.1 The comb function.- 6.2.2 Sampling.- 6.2.3 Interpreting DFT data in a cyclic buffer.- 6.3 The Discrete Fourier Transform.- 6.4 The Fast Fourier Transform.- 6.4.1 The DFT as a matrix equation.- 6.4.2 Simplifying the transition matrix.- 6.4.3 Signal-flow-graph notation.- 6.4.4 The DFT expressed as a signal flow graph.- 6.4.5 Speed advantages of the FFT.- 6.5 Caveats to using the DFT/FFT.- 6.6 Conclusion.- 6.7 Bibliography.

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