Radon Transformation - Digital Signal Processing for Medical Imaging Using Matlab - page 1

Chapter 1
Radon Transformation

1.1 Introduction to Computed Tomography (CT)

The physical setup to obtain the CT of the particular slice of the test body involves
passing the X-ray to that particular slice and detecting the attenuated signal at the
other side. This value is conceptually proportional to the integral value of sliced image
along the X-ray paths. The ray-path directions with respect to sliced image describes
the type of projection used in that CT. There are two major types of projection
techniques namely Parallel beam projection and Fan-beam projection used in CT.
The process of reconstructing the image from the projected data involves digital
signal processing, which are described below.

1.2 Parallel Beam Projection

Let us consider an example image (refer Fig. 1.1 ) as the sliced image of the test
body. The parallel beam projection involves transmitting X-ray signals one after and
another, parallel to each other and the corresponding attenuated signals are captured
using the detector kept exactly on the other sides of the ray (refer Fig. 1.2 ). This is
equivalent to obtaining the line integration of the image in the direction of the parallel
beam. This is the radon transformation with an angle 0 ◦ . Now the image is rotated
in the clock-wise direction with an angle θ ◦ and the line integration is computed

as mentioned above. This is the radon transformation with angle θ ◦ . (In practice,

this is obtained by shifting the positions of the source and the detector such that
the imaginary line joining the source and the detector is rotated in the anticlockwise
direction by an angle

θ ◦ ranging from 0 to 360 ◦ .
This completes the forward radon transformation. The process of estimating the
original image from the forward radon transformation data is called as inverse radon
transformation, which is described below.

θ ◦ ). This is repeated for the angle