sLecture

↳ Topic 2: Tomographic Reconstruction

↳ Intro

↳ CT

↳ PET

Excerpt from Prof. Bouman's Lecture

Accompanying Lecture Notes

Tomography refers to the method of producing images of plane sections, or slices of a solid object. The word is derived from Greek tomos meaning section.

At this point let me introduce a few common medical imaging modalities:

• Anatomical Imaging Modalities
  • Chest X-ray
  • Computed Tomography (CT)
  • Magnetic Resonance Imaging (MRI)
• Functional Imaging Modalities
  • Signal Photon Emission Tomography (SPECT)
  • Positron Emission Tomography (PET)
  • Functional Magnetic Resonance Imaging (fMRI)

Anatomical imaging modalities only reveal the structure of an object, Figure 1, for example, compares MRI scans of two patients. Functional imaging modalities can differentiate between active and inactive cells. So they can sort of tell you what the tissue is doing. Figure 2 shows an fMRI scan of a woman's brain after a stroke.

Fig 1: MRI scans comparing a normal brain (top) to that of a patient suffering from Ataxia (bottom)

Fig 2: fMRI scan of a brain after a stroke. Blues and greens correspond to normal blood flow, reds and blacks correspond to abnormal blood flow.

It is important to realize that CT, PET and MRI are three-dimensional imaging modalities and must be visualized in voxels. Figure 3 illustrates how an MRI scan produces multiple slices to build up a representation of a three-dimensional volume. Compare this to the classical chest X-ray, which is a two-dimensional scan and built up of pixels. When you look at a planar chest X-ray, you see the image of the heart, the lungs, the rib cage and the vertebrae superimposed on one another. This makes accurate diagnosis difficult.

Fig 3: MRI scan of one half of one hemisphere of the brain

We can use the CSFT to analyze applications such as tomography and MRI imaging without getting into sampling and discrete time systems. Of course in reality, these scans are performed by computers and must be implemented digitally but for now the CSFT will suffice in building an understanding of these systems.

While the measurements from MRI and tomography can both be analyzed by CSFT, both the physics and the mathematics behind the two cases are completely different as they are governed by very different sets of equations that represent two very distinct techniques.

CT and PET are two examples of modalities that use tomography to reconstruct an image. While the physics of these two modalities are very different, we will see that for both cases, we measure the line integral of the density through the material. So if you get the projection through the object at each point, at every angle $ \theta $, you get the one-dimensional image of the projections. Once you have made the measurement form every angle, you can reconstruct the object. It's as if you are looking at an object and you cannot see around it because the object blocks your view but if you tilt your head to different angles, you can see the back of the object. So that is the intuition behind tomography, that by looking at different angles, you can obtain different views and form a consistent interpretation of all the views. If you have only a single view, there is information you have lost. But if you have lots of different views, you can put them all together to get a three-dimensional interpretation of the object. This is similar to how both your eyes work together in binocular vision to give you a perception of depth. You have two points of view and your brain processes the two separate images to allow you to perceive your surroundings in three dimensions.

later we'll see that tomographic reconstruction is an example of an inversion process where your measurements are the linear transform of your object and you are trying to invert the linear transform.