There are, of course, many other medical imaging modalities. Ultrasound is a real-time and largely 2D modality, but can be made 3D by the addition of tracking or 2D phased array probes. There is nuclear medicine, SPECT, thermography, electrical impedance tomography, elastography, optical coherence tomography, confocal microscopy, magnetoencephalography, fluorescence lifetime imaging, near infrared optical tomography and spectroscopy and the list goes on. It is not possible to cover the huge field of medical imaging in one chapter. Rather, we provide an introduction to the subject that summarizes the basics of 3D medical imaging. For more details, we refer the reader to some excellent textbooks [4, 16, 39], as well as online references; for example, the online Encyclopedia of Medical Physics (EMITEL) [77].

11.2 Volumetric Data Acquisition

Before discussing how we can process and analyze 3D medical imaging data, it is important to examine the issues relating to data acquisition. The human body is largely opaque to optical imaging, so medical images must use other physical processes to image tissue properties. The techniques used will have significant influence on the types of tissue that can be imaged and the quality of the 3D data obtained in terms of contrast, noise characteristics and artifacts.

In this section, we will summarize the methods behind volumetric data acquisition. For 3D imaging of human anatomy, two modalities dominate, CT and MRI. We describe briefly the physics and the computational methods used to reconstruct these modalities, as well as considering the functional modality, PET. We consider the characteristics of the reconstructed data from the point-of-view of 3D and describe some of the artifacts that can occur.

11.2.1 Computed Tomography

Computed tomography (CT) is essentially a 3D version of classical X-Rays. The overall idea is fairly simple. An X-Ray is a projection through the object, in the sense that the pixel intensities can be interpreted as related to an integral along projected rays through the object. If we take multiple X-Rays at different angles, we might be able to solve for the 3D voxel intensities. In fact, the process really reconstructs a single 2D slice and the patient is moved through the scanning plane to collect multiple slices and create a 3D volume.

So, how is such a 2D slice reconstructed? The object (slice through the patient) can be considered as a 2D array of X-ray attenuation coefficients μ(x, y). The aim of CT imaging is then to reconstruct the function μ(x, y). If the incident intensity from the X-ray source is I0, the transmitted intensity I having passed along a single ray through the patient will be:

where (x(s, θ ), y(s, θ )) denotes a point along the ray making an angle θ with an arbitrary but fixed axis, and s is distance along the ray. We can convert this to a direct integral along the line by taking log intensities, in which case the process corresponds mathematically to a Radon transform. We are integrating the function μ(x, y) along a given line. If we have a parallel beam geometry, many such lines are integrated through the object at the same angle, θ , but different offsets, d , providing a linear projection profile, P , of the object. The 1D projection at this angle becomes:

This corresponds to the Radon transform, but to reconstruct μ(x, y) we must solve for the inverse Radon transform. There is a nice theorem that helps in this process, known as the central slice theorem or the projection slice theorem. It turns out that the 1D Fourier transform of the projection P is actually the same as a line through the 2D Fourier transform of μ(x, y). Hence, we can take the 1D Fourier transform of all our projections at different angles and use these to interpolate the full 2D Fourier transform. The final image is then obtained by a 2D inverse Fourier transform of the result.

A simpler approach is merely to back-project (spread the value associated to a line uniformly among all pixels on the line) the profiles at all angles and average them. This produces images which are rather blurred and this blurring has an analytical expression. Correct results are achieved if the individual slice spectra are filtered before undergoing a 1D inverse Fourier transform and then being projected back into the 2D spatial image. Each filtered projection is simply projected back into the 2D image at the same angle as it was taken and the sum of all projections is then averaged. This process is known as filtered back-projection. An excellent online demo is available [45].

We have described how a 2D slice is produced. By incrementing the slice position, we end up with a stack of slices corresponding to the full 3D volume. In modern scanners, the reconstruction is not from parallel beams, but fan beams (single source to multiple detectors in a ring). The acquisition is often taken using a spiral motion with the patient continually moving through the scanner and reconstruction being fully 3D. An example CT scan is shown in Fig. 11.1. Multiple slices are often detected in one acquisition and the latest scanners may be 256 or even 512 slice devices. This enables, for example, coronary scans to be taken in a single heartbeat.

11.2.1.1 Characteristics of 3D CT Data

Since CT images the X-ray attenuation coefficient, it is good at imaging dense objects such as bones. The contrast for soft tissues may not be so high, but certainly fat and muscle can be differentiated and often pathological tissue such as cancerous lesions can be identified. The intensity corresponding to a given tissue density should always be the same and CT voxels are generally provided in Hounsfield units

Fig. 11.1 Slices from an abdominal CT and volume rendering showing pelvic bone and major vessels

(HU) (−1000 for air, 0 for water) or CT number (HU + 1000). Hounsfield units are named after Sir Godfrey Hounsfield, who received the Nobel Prize for Medicine in 1979 for his pioneering work in constructing the first CT scanner.

The resolution of a CT slice is generally less than 1 mm. Image dimensions are typically a power of 2 and in-slice matrix will typically be 512 × 512. The thickness and separation of slices varies depending on the application, but will typically be a few millimeters unless there is a specific desire for an accurate 3D model. Since CT consists of multiple X-rays, radiation dose is a major issue. As with X-rays, there is a compromise between image quality and dose. The relationship between dose, D, and signal-to-noise ratio (SNR) can be expressed as:

D SNR2 d3h

where d is in-plane pixel size, and h is slice thickness [16]. At the same time there is a pay-off between resolution and SNR. Greater resolution, whether in-plane or in terms of slice thickness, will mean a reduction in SNR.

There are many artifacts that can arise in CT scanning. If very dense objects, such as metal fillings, are visible in the scan, the back-projection process will mean that these appear as streaks in the reconstructed image [37]. If there is a region of high density material, for example a thick area of skull bone, this effectively filters out lower energy X-rays and leaves a higher energy beam that is less easily absorbed. The result can be shadows beyond such regions and these are known as beam hardening artifacts. Motion can also cause significant artifacts in CT (see Fig. 11.2).

Despite the potential for artifacts and radiation dose, CT is able to produce some spectacular anatomical reconstructions (see Fig. 11.1). It is also worth noting that