Image or a slice of a 3-D object is broken down into a set of 1-D projections Each projection is filtered individually These projections are backprojected (smeared) together Original image or cross-section reconstructed
Introduction
An important problem in digital imaging today exists in the reconstruction of 3-D images from numerous 2-D cross-sectional samples of that original 3-D image. Those 2-D cross-sections are created in computed tomography (CT) which entails the recontruction of an image from a sum of projections, each taken at a different angle around the image. These projections refer to the mapping of an image into a waveform whose values are sums of line integrals in different directions. This gives us a set of 1-D projections which when summed, make up the original 2-D cross-sectional image. Applications for tomography range from medicine to science to technology with specific areas such as: diagnostic radiology, Single Photon Emission Computerized Tomography (SPECT), Position Emission Tomography (PET), Magnetic Resonance Imaging (MRI), Synthetic Aperture Radar (SAR), radioastronomy, optical interferometry, and quantum optics. The list goes on and on. In medicine, there are alternatives to tomography, however, oftentimes they are very painful and sometimes life threatening techniques.
Due to some limitations (it is difficult for us to cart around an MRI machine), we implemented our entire project in MATLAB. In addition, we will not take cross-sections of 3-D images rather we will take 2-D objects and reconstruct them with a sum of 1-D projections. For our gang project, we aim to answer 3 questions:
At what point during the backprojection does the sum of the smeared projections start to resemble the original 2-D image?
What type of filter yields the most accurate recreation of our image?
Are there any situations/circumstances where a different filter produces more desirable results?
To preview the methods performed for this project, the basic process behind Tomography is broken down into a few easy steps:
Image or a slice of a 3-D object is broken down into a set of 1-D projections Each projection is filtered individually These projections are backprojected (smeared) together Original image or cross-section reconstructed
The rest of this report will aim at elaborating on the above process. To begin with, a background in the mathematical theory behind tomography will be explained. Then, the ideas and goals of this project will be elaborated on and the results from our individual experiments will be provided. Finally, we will try to move beyond just theoretical ideas and attempt to simulate a real-world environment and try to understand the problems facing scientists and doctors dealing with tomography in their everyday jobs.
