Detailed descriptions of various tomographic imaging techniques are offered in this book. It provides mathematical details that are suitable for graduate students, practicing engineers, and scientists in the field of medical imaging.

The content can be divided into three parts: fundamentals, traditional transmission tomography, and diffraction/reflection tomography. The part on fundamentals covers necessary mathematics and signal processing material for completeness, such as Radon Transform, Fourier Transform, and Fast Fourier Transform. The second part covers traditional techniques such as CT scans, MRI, Ultrasound, PET and SPET. The third part covers less familiar techniques for situations where diffraction and reflection of source signals cannot be ignored. The contents of the first part can be found in regular image processing textbooks [1,2]. The second part can be also found in these books, although in less detail. These chapters serve an excellent reference for practitioners. The third part consists of three chapters (6 to 8), and covers more advanced topics that involve diffraction and/or reflection of source signals. These topics are important for seismology and medical applications, and the book provides the fundamentals of relevant techniques in a concise manner. These chapters distinguish Kak and Slaney’s work from other similar books, and make it valuable for researchers.

Chapter 6 describes tomography with diffracted waves. The Fourier slice theorem is no longer applicable in these situations. The theory needs to be amended, and this chapter shows that the projection of diffracted waves in the Fourier domain corresponds to the Fourier transform along a semi-circular arc of a scalar function defined with the object’s magnetic permeability and dielectric constant. The theorem is based on an approximation of scatter fields using either Born or Rytov approximation; the approximations and thus the theorem are valid for a small object with small refractive index. Major issues of reconstruction using the Fourier diffraction theorem include how to cover whole frequency domains efficiently with a finite number of semi-circular arcs, and how to interpolate values accurately at a regular rectangular lattice. For the non-diffraction case, the issues are simpler. Three techniques discussed in the book are polynomial interpolation in the frequency domain, unified frequency domain reconstruction, and filtered backpropagation. Pros and cons of each technique are described. The chapter concludes with an extensive amount of simulation and numerical performance analysis of the reconstruction algorithms.

Chapter 7 describes algebraic techniques. The technique treats the reconstruction problem as an algebraic solution of simultaneous linear equations; thus no Fourier transformation is involved. The technique is known to be less accurate and computationally more expensive than the transformation based techniques. However, it becomes the method of choice when a large number of projections or uniformly distributed projections are not available. Computational efficiency and numerical stability are keys to the technique. This chapter introduces three techniques: algebraic reconstruction, simultaneous iterative reconstruction, and simultaneous algebraic reconstruction. Extensive simulation experiments given in the chapter evaluate the methods in terms of computational load and reconstruction accuracy.

Finally, chapter 8 describes tomography of reflecting sources where the measurement is applied to the reflected returns of the input pulses. The technique is common in seismology, where the transmitter and receiver have to be placed on the same side of the object, and in medical ultrasonic imaging, where large impedance discontinuities at tissue-bone and air-tissue boundaries make conventional transmission tomography impractical. With this technique, short pulses are transmitted, and their returns are observed over time. One can formulate the relationship between the temporal profiles of the return signal with the reflectivity distribution of the object along the direction of the transmitted pulse. With a number of pulses at different directions and corresponding returns, the reflectivity distribution of the entire object can be reconstructed.

This is a reprint of the book originally published in 1987. Very little has been added to this version since the original publication, and the authors suggest that readers complement the book with more recent publications. Despite this shortcoming, the book remains very helpful to those involved in tomographic imaging.