The Fourier Transform in Biomedical Engineering / Edition 1

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Overview

The Fourier Transform in Biomedical Engineering surveys and introduces the Fourier Transform as a useful and practical technique in the biomedical sciences. The material covered in this primer will educate the reader in the fundamentals of Fourier theory through the use of practical demonstrations and examples. This new text is intended to interest scientists in biomedicine, medical imaging, and related fields. The careful explanations and many examples provide the essential material for those who require an understanding of the significance of temporal and spatial spectral analysis. This book will also serve as a useful resource to medical technologists and physicians for whom a grasp of Fourier principles would aid in their understanding of instruments such as CT and MR scanners.

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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