Linear algebra, urgent assignment.

A single x-ray passing through a body is absorbed at rates depending upon the material it goes through. Thus if an x-ray is passed through bone, a certain amount of intensity units is absorbed, while if it passes through soft material, a different amount of intensity units is absorbed. A CT scan is a scan in which an x-ray is passed through an object (part of a body) from multiple angles (on the order of 360), to produce a detailed, high quality two-dimensional image. How does it work? Consider Figure 1, consisting of a toy two-dimensional figure consisting of 16 pixels. Each pixel can absorb 1 intensity unit (i.u.) from the X-ray beams, or none. Let’s assume the white boxes are bone and absorb 1 i.u., and the black pixels absorb no i.u.’s and represent e.g. soft tissue. Let µ1 be the attenuation coefficient of element 1 (denoted by a 1 with a circle around it), µ2 be the attenuation coefficient of element 2, … µ16 be the attenuation coefficient of element 16. Ideally, each µ is either 0 or 1, indicating soft tissue or bone, respectively. A beam from the x-ray can be sent through the material at any location and from any direction, but must start on the outside of the material, and then the amount absorbed is measured where the x-ray exits. The goal is to use x-ray data to reconstruct this figure and determine the vector µ in IR16 . We have 16 pixels so in order to determine the attenuation coefficient of each pixel we need at least 16 equations. Let’s assume we have 16 X-ray measurements, denoted by arrows 1 through 16. Arrow 1 (first column down) would have 1 i.u. absorbed for each white pixel, so the number of i.u.’s coming out would be 4 (one for each white square). Of course a real X-ray would not be this simple, but this gives us an idea of how it works. So the first equation says that if we pass an X-ray through the first column, it goes through pixels 1, 5, 9, and 13, and the number of i.u.’s absorbed is 4. Our first equation is thus:

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