Chapter 3:
Image Reconstruction
3.1 Introduction As discussed in Chapter 2, the object that we try to reconstruct can be considered a 2D distribution of some kind of function. For CT, this function represents the linear attenuation coefficients of the object. The problem imposed on the tomographic reconstruction can be stated as the following: Suppose we have collected a set of measurements, and each measurement represents the summation or line integral of the attenuation coefficients of the object along a particular ray path. These measurements are collected along different angles and different distances from the iso-center. To avoid redundancy in the data sampling, assume that the measurements are taken in the following sequence. First, we take a set of measurements along parallel paths that are uniformly spaced, as shown by the solid lines in Fig. 3.1. These measurements form a “view” or a “projection.” Then we repeat the same measurement process at a slightly different angle, as shown by the dotted lines in Fig. 3.1. This process continues until the entire 360 deg (theoretically, only 180 deg of parallel projections are necessary) is covered. During the entire process, the angular increment between adjacent views remains constant, and the scanned object remains stationary. For CT reconstruction, the question is how to estimate the attenuation distribution of the scanned object based on these measurements. Before we begin rigorous discussions on the mathematical principles of CT, we will present a simple “thinking” experiment to demonstrate how CT could work. Assume that we try to guess the internal structure of a semitransparent object. The object is formed with five spheres embedded inside a cylinder, as shown in Fig. 3.2(a). We are not allowed to view the object in a top-down fashion (looking directly into the page); we can only examine it from the side. If we examine the object only at the angle shown in the figure, two of the spheres are partially blocked (since they are semitransparent) by the sphere in front, and we see only one overlapped sphere. Although the opacity of the overlapped sphere is expected to be higher than the nonoverlapped spheres, we cannot infer from the opacity the number of spheres that overlap, because we do not know the opacity of each sphere. Based on a single view, we may erroneously conclude that the cylinder contains only three spheres.
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