Diagnostic Ultrasound Imaging: Inside Out (eBook)

Thomas L. Szabo (Autor)

eBook Download: PDF | EPUB

2013 | 2. Auflage
832 Seiten
Elsevier Reference Monographs (Verlag)
978-0-12-396542-4 (ISBN)

Infos

Diagnostic Ultrasound Imaging provides a unified description of the physical principles of ultrasound imaging, signal processing, systems and measurements. This comprehensive reference is a core resource for both graduate students and engineers in medical ultrasound research and design. With continuing rapid technological development of ultrasound in medical diagnosis, it is a critical subject for biomedical engineers, clinical and healthcare engineers and practitioners, medical physicists, and related professionals in the fields of signal and image processing. The book contains 17 new and updated chapters covering the fundamentals and latest advances in the area, and includes four appendices, 450 figures (60 available in color on the companion website), and almost 1,500 references. In addition to the continual influx of readers entering the field of ultrasound worldwide who need the broad grounding in the core technologies of ultrasound, this book provides those already working in these areas with clear and comprehensive expositions of these key new topics as well as introductions to state-of-the-art innovations in this field. - Enables practicing engineers, students and clinical professionals to understand the essential physics and signal processing techniques behind modern imaging systems as well as introducing the latest developments that will shape medical ultrasound in the future - Suitable for both newcomers and experienced readers, the practical, progressively organized applied approach is supported by hands-on MATLAB© code and worked examples that enable readers to understand the principles underlying diagnostic and therapeutic ultrasound - Covers the new important developments in the use of medical ultrasound: elastography and high-intensity therapeutic ultrasound. Many new developments are comprehensively reviewed and explained, including aberration correction, acoustic measurements, acoustic radiation force imaging, alternate imaging architectures, bioeffects: diagnostic to therapeutic, Fourier transform imaging, multimode imaging, plane wave compounding, research platforms, synthetic aperture, vector Doppler, transient shear wave elastography, ultrafast imaging and Doppler, functional ultrasound and viscoelastic models

Professor Szabo has contributed to the fundamental understanding and design of surface acoustic wave signal processing devices, to novel means of transduction and measurement for nondestructive evaluation using ultrasound, to seismic signal processing, and to the research and development of state-of-the-art diagnostic ultrasound imaging systems for over fifty years. He is the author of the widely used textbook, Diagnostic Ultrasound Imaging: Inside Out, over 100 papers and twelve book chapters, and holds four patents and several patent applications. His wide range of interests include ultrasound tissue and spine characterization, wave equations, novel imaging systems, brain imaging, therapeutic ultrasound, nonlinear phenomena and geophysical exploration. Dr. Szabo is a Fellow of the American Institute of Ultrasound in Medicine, Acoustical Society of America, and a Life Senior member of the IEEE. He is a U.S. delegate to the International Electrotechnical Commission (IEC), Technical Committee 87 and a Convenor of Working Group 6 on high intensity therapeutic ultrasound and focusing. He was a recipient of a 1973 U. S. Meritorious Service Medal, a Hewlett Packard Fellowship and the 1974 best paper award in the IEEE Transactions on Sonics and Ultrasonics.

Textauszug

Chapter 2

Overview

A block diagram approach provides a way of organizing the functions of various physical processes and signal and imaging processing of a diagnostic ultrasound imaging system. This chapter introduces the overall block central diagram structure that links and introduces upcoming chapters, each of which explains the principles of blocks in more detail. Time and frequency −i Fourier transforms are used to describe changes along the signal chain and space, and (wave number) spatial frequency +i Fourier transforms are employed to diffraction and scattering effects. Examples are provided in MATLAB programs.

Keywords

Fourier transform; building block; central diagram; spatial transform; line source; sinc function; scaling theorem; Gaussian; rect function; impulse; tone burst; fast Fourier transform; discrete Fourier transform

Chapter Outline

2.1 Introduction

2.2 Fourier Transform

2.2.1 Introduction to the Fourier Transform

2.2.2 Fourier Transform Relationships

2.3 Building Blocks

2.3.1 Time and Frequency Building Blocks

2.3.2 Space Wave Number Building Block

2.4 Central Diagram

References

2.1 Introduction

Ultrasound imaging is a complicated interplay between physical principles and signal-processing methods, so it provides many opportunities to apply acoustic and signal-processing principles to relevant and interesting problems. In order to better explain the workings of the overall imaging process, this book uses a block diagram approach to organize various parts, their functions, and their physical processes. Building blocks reduce a complex structure to understandable pieces. This chapter introduces the overall organization that links upcoming chapters, each of which describes the principles of blocks in more detail. The next sections identify the principles used to relate the building blocks to each other and apply MATLAB programs to illustrate concepts.

2.2 Fourier Transform

2.2.1 Introduction to the Fourier Transform

Signals such as the Gaussian pulse in Figure 2.1 can be represented as either a time waveform or as a complex spectrum that has both magnitude and phase. These forms are alternate but completely equivalent ways of describing the same pulse. Some problems are more easily solved in the frequency domain, while others are better done in the time domain. Consequently, it will be necessary to use a method to switch from one domain to another. Joseph Fourier, a nineteenth century French mathematician, had an important insight that a waveform repeating in time could be synthesized from a sum of simple sines and cosines of different frequencies and phases (Bracewell, 2000). These frequencies are harmonically related by integers: a fundamental frequency (f0) and its harmonics, which are integral multiples (2f0, 3f0, etc.). This sum forms the famous Fourier series.

Figure 2.1 Forms of the Gaussian pulse. (A) Short 5-MHz time pulse and its (B) spectrum magnitude and phase.

While the Fourier series is interesting from a historical point of view and its applicability to certain types of problems, there is a much more convenient way of doing Fourier analysis. A continuous spectrum can be obtained from a time waveform through a single mathematical operation called the “Fourier transform.” The minus i Fourier transform, also known as the Fourier integral, is defined as:

(2.1)

in which H(f) (with an upper-case letter convention for the transform) is the minus i Fourier transform of h(t) (lower-case letter for the function), “i” is , and symbolizes the minus i Fourier transform operator. Note that, in general, both h(t) and H(f) may be complex, with both real and imaginary parts. Another operation, the minus i inverse Fourier transform, can be used to recover h(t) from H(f) as follows:

(2.2)

In this equation, is the symbol for the inverse minus i Fourier transform. A sufficient but not necessary condition for a Fourier transform is the existence of the absolute value of the function over the same infinite limits; another condition is a finite number of discontinuities in the function to be transformed. If a function is physically realizable, it most likely will have a transform. Certain generalized functions that exist in a limiting sense and that may represent measurement extremes (such as an impulse in time or a pure tone) are convenient and useful abstractions. The Fourier transform also provides an elegant and powerful way of calculating a sequence of operations represented by a series of building blocks, as shown shortly.

For applications involving a sequence of numbers or data, a more appropriate form of the Fourier transform, the discrete Fourier transform (DFT), has been devised. The DFT consists of a discrete sum of N-weighted complex exponents, exp(−i2π mn/N), in which m and n are integers. J. W. Cooley and J. W. Tukey (1965) introduced an efficient way of calculating the DFT called the fast Fourier transform (FFT). The DFT and its inverse are now routine mathematical algorithms and have been implemented directly into signal processing chips.

2.2.2 Fourier Transform Relationships

The most important relationships for the Fourier transform, the DFT, and their application are reviewed in Appendix A. This section emphasizes only key features of the Fourier transform, but additional references are provided for more background and details.

A key Fourier transform relationship is that time lengths and frequency lengths are related reciprocally. A short time pulse has a wide extent in frequency, or a broad bandwidth. Similarly, a long pulse, such as a tone burst of n cycles, has a narrow band spectrum. These pulses are illustrated in Figure 2.2 and Figure 2.3. If, for example, a tone burst of 10 cycles in Figure 2.2 is halved to 5 cycles in Figure 2.3, its spectrum is doubled in width. All of these effects can be explained mathematically by the Fourier transform scaling theorem:

(2.3)

For this example, if g(t) is shown in Figure 2.2, then for the shorter length signal in Figure 2.3, if a=2.0, then the spectrum is halved in amplitude and its width is stretched by a factor of two in its frequency extent. Many other Fourier transform theorems are listed in Table A.1 of Appendix A.

Figure 2.2 A 5-MHz center frequency tone burst of 10 cycles and its spectral magnitude.

Figure 2.3 A 5-MHz center frequency tone burst of 5 cycles and its spectral magnitude.

Consider the Fourier transform pair from this table for a Gaussian function:

(2.4)

To find the minus i Fourier transform of a following given time domain Gaussian analytically, for example,

(2.5A)

first put it into a form appropriate for the scaling theorem, Eqn 2.3, and the Gaussian, Eqn 2.4,

(2.5B)

so that Then by the scaling theorem, the transform is:

(2.6)

The Gaussian is well behaved and has smooth time and frequency transitions. Fast time transitions have a wide spectral extent. An extreme example of this characteristic is the impulse in Figure 2.4. This pulse is so short in time that, in practical terms, it appears as a spike or as a signal amplitude occurring only at the smallest measurable time increment. The ideal impulse would have a flat spectrum (or an extremely wide one in realistic terms). The converse of the impulse in time is a tone burst so long that it would mimic a sine wave as in Figure 2.5. The spectrum of this nearly pure tone would appear on a spectrum analyzer (an instrument for measuring the spectra of signals) as either an amplitude at a single frequency in the smallest resolvable frequency resolution cell or as a spectral impulse. Note that instead of a pair of spectral lines representing impulse functions in Figure 2.5, finite width spectra are shown as a consequence of the finite length time waveform used for this calculation by a digital Fourier transform. All of these effects can be demonstrated beautifully by the Fourier transform. The Fourier transform operations for Figures 2.1–2.5 were implemented by MATLAB program chap2figs.m.

Figure 2.4 A time impulse and its spectral magnitude.

Figure 2.5 A 5-MHz pure tone and its spectral magnitude.

2.3 Building Blocks

2.3.1 Time and Frequency Building Blocks

One of the motivations for using the Fourier transform is that it can describe how a signal changes its form as it propagates or when it is sent through a device or filter. Both of these changes can be represented by a building block. Assume there is a filter that has a time response, q(t), and a frequency response, Q(f). Each of these responses can be represented by a building...

Produkt-Details

• Erscheint lt. Verlag: 5.12.2013
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Medizin / Pharmazie ► Gesundheitsfachberufe
• Themenwelt: Medizinische Fachgebiete ► Radiologie / Bildgebende Verfahren ► Sonographie / Echokardiographie
• Themenwelt: Medizin / Pharmazie ► Physiotherapie / Ergotherapie ► Orthopädie
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Angewandte Physik
• Themenwelt: Technik ► Medizintechnik
• ISBN-10: 0-12-396542-X / 012396542X
• ISBN-13: 978-0-12-396542-4 / 9780123965424