Sample Data for Problem 1a A =[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 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 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ] MATH 220 - CT Scan Project (in class) Directions This .

1. Sample Data for Problem 1a
A =[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
1 0 0 0 0 1 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
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
0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0

2. 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
1 0 0 0 0 1 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0
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
0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

3. 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ]
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

4. 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,

5. 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.
16
1 2
3 4
5
8

6. 7
6
9
10
11
12
13 14 1516 17 18 19
20
21
22
1 3 4
5 7 8
9 10 12
13 14 15
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 mea-

7. surements 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
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 4
0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 2
0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 3
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 4
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 3
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 3
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 3
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 4
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 3
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2
0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 2
0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
.

8. 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:
ATAµ = ATb

9. 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 equa-
tions?
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”

10. 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.
24
1 2
3
3 4 5 621
4 5 6
7
8
9
7 8 9 10 11 12
181716151413
10
12

11. 11
13 14
15
16 1718
19 20 21 22 23
25
Figure 2: System for Individual Part of Project
Possibly Helpful Matlab Commands
rank(A) Gives the rank of the square matrix A
A’ Gives the transpose of the matrix A, AT
A*b Multiplies matrix A and matrix b, Ab
det(A) Gives the determinant of A, det(A)
x=Ab Solves Ax = b using row reduction and stores the answer,
x, into the vector x.