Radiography and Tomography

Possible Course Linear Algebra

Application The motivating application for this module is computerized tomography (CT)imaging. In this procedure, multiple images are taken of a 3D object, and the collection of theseimages forms the scan. Students will be given radiographic data from an MRI brain scan andwill be tasked with reconstructing enough information about the brain to determine the possiblepresence of some anomalyy.

Motivated Concepts Vector spaces, span, basis, linear transformations, null space, columnspace, the rank theorem, diagonalization, injectivity, surjectivity, invertibility, orthogonal matrices,pseudoinverse and matrix factorization.

Prerequisite Material Vector Calculus is useful for visualization. Other prerequisites and mate-rial presentation suggestions are described individually for each Lab.

Description In this module students will be introduced to radiography and tomography as phys-ical applications of linear algebra. The module will begin by introducing the space of images asa vector space. Students will explore the mathematics behind taking radiographs, or imaging anobject onto a lower-dimensional subspace. (The imaging is often a projection map.) They will alsoconsider the inverse: given the radiograph, can one “reconstruct” the original object? Many earlyconcepts – such as null space, span, basis, range – appear naturally as a consequence of exploringproperties of objects and their radiographs.

When an operator is not invertible, students will explore the existence of a one-sided inverseand a “pseudo-inverse” operator obtained through the singular value decomposition of a matrix.They will use this to provide approximate reconstructions of objects. They will also extend theideas developed in the module to creatively explore how to leverage prior information about theobject and how to deal with noisy data.

The module consists of between 4 and 7 computer labs, and will scalable so that students canstill arrive at interesting reconstruction results even if the class will not be able to include thesingular value decomposition.

Time Frame The labs in this module would probably take around 2.5-5 weeks of class time,depending on how much of the material instructors will incorporate. They would be interleavedthroughout the course. In conjuction with the Heat Diffusion Module, this module inspires most ofthe key concepts in a first course in Linear Algebra and many concepts found in a second course.

Labs 1 through 5 constitute the base material leading up to the students finding their firstobject reconstructions from radiographic data. In the process, they discover many of the keyconcepts pertaining to vector spaces and linear transformations. Labs 6 and 7 develop advancedconcepts associated with singular value decomposition and allow the students to obtain dramaticallyimproved object reconstructions.

Lab 1 Students explore the idea of an image as a member of a set of objects and how one naturallyperorms arithmetic on images. This exploration motivates ideas of vector, vector space, linearcombination, span and properties of a vector space. No prior material from linear algebrais required. This lab would make a good introduction to vector spaces, or could come rightafter vector spaces have been defined.

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Lab 2 Students will explore the properties of linear transformations through constructing and usingsmall examples of radiographic operators. Students should have a brief introduction to thephysics of radiography before this lab. A summary document is provided for this purpose assupplementary reading. They should be very comfortable with matrix-vector multiplication.

Lab 3 This lab introduces injectivity, surjectivity and the null space of linear transformations in thecontext of radiographs. Students should not have proved theorems in this area yet, since themain idea is to get them to explore and discover some of the key ideas surrounding theseconcepts. This lab leads very naturally to the Rank Theorem.

Lab 4 Students investigate the invertibility of a radiographic transformation. This lab can be doneas a self-directed exercise or homework if the class has already discussed invertible transfor-mations. It can also serve as a review of important concepts just before Lab 5.

Lab 5 In this lab students explore one way in which they might go about reconstructing the orig-inal 3D object from a radiograph when the radiographic transformation is not invertible.Specifically, they create a left-inverse of the radiographic transformation. This motivatesone-sided inverses. In their reconstruction, they may also notice some flaws inherent to theprocess, which can be used as motivation for the use of the singular value decomposition inthe next lab. Students should be already familiar with properties of symmetric matrices andrelationships between rank, column space and null space of a matrix and its transpose.

Lab 6 Students will explore pseudoinverses obtained through the singular value decomposistion(SVD) of a transformation. Pseudoinverses can be radiograph-noise tolerant and exist fornon-injective transformations. Students should be already familiar with diagonalization andorthogonal matrices.

Lab 7 Students will explore the use of the null space of a radiographic operator to enhance recon-structions based on object prior knowledge. This method can lead to dramatic improvementsin reconstructions from limited radiographic data. Prerequisite material includes a firm under-standing of Lab 6 and working with projections onto subspaces.

Successful use of this module does not require completion of all labs. Possible paths throughthe module include completing the first four Labs and then finishing with any combination of labs5, 6, and 7.

Figure 1: Example radiographs of a human head.

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Figure 2: Example 2d slices of a 3d reconstruction.

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