18.1 INTRODUCTION
In the second part of computerized imaging involving Fourier-related transforms,
image reconstruction from projections including tomography is discussed. The
fundamental transform for this purpose is the Radon transform. In this chapter, the
Radon transform and its inverse are first described in detail, followed by imaging
algorithms used in tomography and other related areas.
Computed tomography (CT) is mostly used as a medical imaging technique
in which an area of the subject’s body that is not externally visible is investigated. A
3-D image of the object is obtained from a large series of 2-D x-ray measurements.
An example of CT image is shown in Figure 18.1. CT is also used in other fields such
as nondestructive materials testing.
This chapter consists of eight sections. The Radon transform is introduced in
Section 18.2. The projection slice theorem that shows how the 1-D Fourier
transforms of projections are equivalent to the slices in the 2-D Fourier transform
plane of the image is discussed in Section 18.3. The inverse Radon transform (IRT)
is covered in Section 18.4. The properties of the Radon transform are described in
Section 18.5.
The remaining sections are on the reconstruction algorithms. Section 18.6
illustrates the sampling issues involved for the reconstruction of a 2-D signal from
its projections. Section 18.7 covers the Fourier reconstruction algorithm, and
Section 18.8 describes the filtered backprojection algorithm.
TABLE OF CONTENTS Projecting images in radar and medical applications
From DSP-FPGA.com
An FPGA contributes to a reconfigurable system’s performance by allowing a programmer to explicitly and completely dedicate a device to the solution of the regular, uninterrupted streaming aspects of a program.
Examples of computationally intense systems being shrunk by use of FPGAs are of keen interest to us. Filtered backprojection is finding its way into both radar and medical imaging applications, and is well served by FPGAs to handle a portion of the algorithm. The results are outstanding. Searching for an enemy vehicle from a high-flying Unmanned Aerial Vehicle (UAV). Taking a child after a tumble from a bike to the hospital for a CT scan of their broken arm. The imaging techniques in these two seemingly unrelated scenarios rely on the same technology to produce accurate results: the Filtered Backprojection (FBP) algorithm.
Searching for an enemy vehicle from a high-flying Unmanned Aerial Vehicle (UAV). Taking a child after a tumble from a bike to the hospital for a CT scan of their broken arm. The imaging techniques in these two seemingly unrelated scenarios rely on the same technology to produce accurate results: the Filtered Backprojection (FBP) algorithm.
The FBP algorithm reconstructs an image of an object by calculating an exact solution for each pixel from a series of planar image data sets (projections) of the object. Both radar and medical imaging require a high degree of imaging accuracy. The FBP algorithm provides excellent imaging accuracy but at a large computational cost.
Heterogeneous reconfigurable systems provide the required computational power for image reconstruction to both Synthetic Aperture Radar (SAR) Backprojection imaging and Computed Tomography (CT) imaging while delivering such significant benefits as smaller form factors and reduced power consumption.
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