Unformatted Attachment Preview
Sample Data for Problem 1a
A =[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
0100010001000100
0010001000100010
0001000100010001
111 1000000000000
0000111 100000000
0000000011110000
0000000000001111
0000000000000100
0000000010000100
0000100001000010
1000010000100001
0100001000010000
0010000100000000
0001000000000000
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
b=[4 2 3 4 3 3 3 4 1 2 3 2 2 2 1 1]
Sample data for Problem 2a. Use your modified matrix A from problem 1.
b=[4.05 2.1 2.9 3.95 2.9 3.05 3.1 3.95 1.1 1.95 3.1 1.9 1.95 2.1 1.05 0.95 1.1 1.95]
Sample data for Problem 2b.
A =[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
0100010001000100
0010001000100010
0001000100010001
111 1000000000000
0000111 100000000
0000000011110000
0000000000001111
0000000000000100
0000000010000100
0000100001000010
1000010000100001
0100001000010000
0010000100000000
0001000000000000
1000000000000000
0100100000000000
0010010010000000
0001001001001000
0000000100100100
0000000000010010
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]
b=[4.05 2.1 2.9 3.95 2.9 3.05 3.1 3.95 1.1 1.95 3.1 1.9 1.95 2.1 1.05 0.95 1.1 1.95 4.03 1.92 2.06
0.9]
Matrix A for individual Projects, problem 3.
A=[1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0
0100000100 00010000
0010000010 00001000
0001000001 00000100
0000100000 10000010
0000010000 01000001
1111110000 00000000
0000001111 11000000
0000000000 00111111
0000000000 00100000
0000001000 00010000
1000000100 00001000
0100000010 00000100
0010000001 00000010
0001000000 10000001
0000100000 01000000
0000010000 00000000
1000000000 00000000
0100001000 00000000
0010000100 00100000
0001000010 00010000
0000100001 00001000
0000010000 10000100
0000000000 01000010
0000000000 00000001 ]
MATH 220 - CT Scan Project
(in class)
Directions This project is due Thursday April 26 at the beginning of class. There are two
parts to this project - the first is an introduction to computed tomography scan, or CT scan, and
works through a sample project while explaining how CT scan images are produced. If there is
time, this will be worked through in class. The second part is an individual project, in which you
are given the output of a two-dimensional CT scan and you are to determine what the picture is.
For this project, we do not ask that you summarize the statement of each problem, nor do
we want you to turn in this paper. Please turn in one sheet of paper with the answers to each
question clearly written. Answer each question using complete sentences. Answers that
simply indicate a single number, a single equation, etc, will be given no credit. Although you may
work in groups or even as a class, your responses should be in your own words. Any indication
of plagiarism, such as duplicate sentences, will be treated as a violation of academic integrity,
resulting in a zero on this project and the dishonesty reported to the Office of Academic Integrity.
This project will be worth 3% of your course grade. Technology allowed. The data is given in a
file that assumes you are using MatLab.
CT Scan Project
The radiation x-ray was discovered by a German physicist, Wilhelm Roentgen, who did not
know what it was, so he simply called it “X-radiation”. 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:
µ1 + µ5 + µ9 + µ13 = 4.
2
16
1 13
14
3
4
17 15 18
19
12
3
1
4
5
11
20
5
8
7
6
10
21
9
10
12
7
9
22
13
14
16
15
8
Figure 1: sample system
Likewise for column 2 we would get the equation:
µ2 + µ6 + µ10 + µ14 = 2.
To solve for 16 unknowns we will need (at least) 16 equations. We have 8 row and column measurements so let’s also use the diagonal measurements 9-16. Diagonal measurements 9 and 10 give
the equations
µ13 = 1
µ9 + µ14 = 2,
respectively. We thus have 16 equations and 16 unknowns giving the following augmented matrix
1
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
4
2
3
4
3
3
3
4
1
2
3
2
2
2
1
1
.
1. Use the data in the file CTScan-Data.docx 1a for the matrix A above and right-hand side, b.
(Note the prime at the end of the b vector - Look at the Matlab commands at the end of the
worksheet to see why it is important to include the prime). Solve the system (sample Matlab
commands are given at the end of this worksheet).
(a) What does Matlab say when you try to solve the system?
(b) Calculate the determinant and rank of A and write your answers.
(c) We could try to eliminate an equation and adding an equation until we got an invertible
matrix, but this is a lot of work. Instead we will try just adding equations. Write the
17th equation using the 17th diagonal sum from Figure 1. Augment your matrix and
right-hand side in your Matlab code.
(d) We cannot calculate the determinant and rank of this matrix, why?
(e) Calculate the RREF of your modified A and write down how many pivots you have.
(f) Continue adding equations until your RREF has 16 pivots (which is equal to the number
of unknowns). Write down the additional equations you used. How many total equations
did you need?
(g) Now solve the equation using least squares:
AT Aµ = AT b
Solving it, you should get the exact solution. Write down your values for µ1 , µ4 , µ6 and
µ16 . Is this consistent with Figure 1? Explain.
2. So this seems easy - what is the catch? The problem with real life, is that experiments are
never exact. So the right-hand side vector b given above may have errors due to measurement.
Use the sample data in CTScan-Data.docx file, 2a for b. Note that the right-hand side is close
to the right-hand side used in problem 1, but with up to 10% error. Solve using least squares.
(a) Write down the values for µ1 , µ6 , µ11 , and µ16 .
(b) Looking at the value for µ16 it is not absolutely clear whether this should be interpreted
as bone or soft tissue. So let us try using all of the equations! Use the data given in
CTScan-Data.docx file, 2b. How many equations and how many unknowns to we have?
(c) Solve the system using least squares. Write down the values for µ1 , µ6 , µ11 , and µ16 .
(d) Compare with the values given using only 18 equations. Is it better to use more equations?
3. Now it is time to do this on your own individual data. In this case we have 18 pixels, see
Figure 2. The matrix associated with this system, consisting of 25 equations and 18 unknowns
is provided in CTScan-Data.docx file, Problem 3. As with the sample problem, the “true”
solution is that each pixel is either white (µ = 1) or black (µ = 0). Your individual output
from the CT scan is provided in a separate file mailed to you. What is the result of your
individual CT scan? The design is in black (µ = 0). It could be a capital letter, a number, a
math symbol, or another design. The designs are oriented so that either elements 13-18 are
on the bottom, or so that elements 1, 7, and 13 are at the bottom. Write down the elements
that you believe are black and explain what it is.
3 15
5
21
6
17 22
23
4
5
6
24
9
10
11
12
25
15
16
17
18
1
2
13 18
1
2
3
7
8
13
14
14 19
4
20 16
12
7
11
8
10
9
Figure 2: System for Individual Part of Project
Possibly Helpful Matlab Commands
rank(A)
A’
A*b
Gives the rank of the square matrix A
Gives the transpose of the matrix A, AT
Multiplies matrix A and matrix b, Ab
det(A)
Gives the determinant of A, det(A)
x=A\b
Solves Ax = b using row reduction and stores the answer, x, into the vector x.