Instructor: Dr. Timothy Feeman :
Department of Mathematics & Statistics, Villanova
University,
800 Lancaster Avenue, Villanova, PA 19085-1699 USA
office: SAC373; phone: (610)519-4693 ;
e-mail link | fax:
(610) 519-6928
Class meetings schedule: MWF 12:30 pm to 1:20 pm in
JBarry 201A.
Final Exam: Saturday, December 17, 2016, from 1:30 pm to 4:00
pm, in JBarry 201A.
CAT scans and MRIs have become commonplace in twenty-first century health care.
Yet the fundamental problem behind these procedures is
completely mathematical:
if we know the values of the integral of a two- or three-dimensional function
along all possible cross-sections, then how can we reconstruct the function
itself? This particular example of what is known as an "inverse problem" was
studied by Radon in the early part of the twentieth century. Radon's work
required a sophisticated use of the theory of transforms and integral operators,
and, by expanding the scope of that theory, contributed to the development of
the rich and vibrant mathematical field of functional analysis. The practical
obstacles to implementing Radon's theories are several. First, Radon's inversion
methods assume knowledge of the behavior of the function along every
cross-section, whilst, in practice, only a discrete set of cross-sections can
feasibly be sampled. Thus, it is possible to construct only an approximation of
the solution. Second, the computational power needed to process a multitude of
discrete measurements and, from them, to obtain a useful approximate solution
has been available for just a few decades. In response to these obstacles, the
past decades have seen a rich and dynamic development both of theoretical
approaches to approximation methods, including the use of interpolation and
filters, and of computer algorithms to effectively implement the approximation
and inversion strategies. Alongside these mathematical and computational
advances, the development, design, and improvement of the machines that actually
perform the tests have also progressed.
This course will focus on the mathematics involved in the creation and analysis of CAT scans. Topics will include Radon and Fourier transforms (both continuous and discrete), convolutions, sampling, filters, and approximate solutions to systems of equations. All of these topics will be discussed in context. Computer projects will explore how to implement basic image reconstruction techniques.
We will use the programming environment R throughout. No prior experience with R is required.
Prerequisites: MAT 2500 and MAT 2705. This course satisfies the "second analysis course" requirement for the Mathematical Sciences major.
The required text for the course is The Mathematics of Medical Imaging: A Beginner's Guide, 2nd edition, by Timothy G. Feeman (Springer, 2015). Several other books have been placed on reserve in Falvey Memorial Library.
R download information: R is an open-source programming environment that we will use in this course for developing simulated medical images, among other applications. My assumption is that you have not had any prior knowledge of R, so we will start from the basics in using it. You can download your personal copy for free here.
Maple download information: We have an unlimited site license for individual users of Maple here at Villanova. That means that you can download your own copy of Maple onto your own computer. This is a great deal that your tuition helps to pay for. So take advantage of it! To get you started, here is a link to the Maple download page on the UNIT Department web site
Homework assignments.
Homework will be assigned regularly.
Selected problems will be collected by me and graded each week.
Remember: "The only way to learn mathematics
is to do mathematics" (Paul Halmos, in
A Hilbert Space Problem Book)
R-based
assignments.
Throughout the course,
you will use the programming environment R to
implement the algorithms that we will be studying and to generate simulated
medical images. I will collect and grade some of this work.
Quizzes and Exams.
We will have
a
10-minute, in-class quiz on most Mondays.
The course
will conclude with a cumulative final exam, to be held on
Saturday, December 17, 2016, from
1:30 pm to 4:00 pm, in JBarry 201A,
as scheduled by the Registrar. You may
bring a 3x5 note card to the final exam.
Your final grade will be based on the following:
Grading: Course grades will not be below what is shown in the following table, where S denotes the weighted score based on the various components.
|
A: 93 ≤ S ≤ 100 |
A-minus: 90 ≤ S < 93 |
B-plus: 85 ≤ S < 90 |
B : 80 ≤ S < 85 |
|
B-minus: 75 ≤ S < 80 |
C-plus: 70 ≤ S < 75 |
C: 60 ≤ S < 70 |
C-minus: 50 ≤ S < 60 |
University
policy statements on disabilities, learning support, and academic integrity:
·
Office of Disabilities and Learning Support Services:
Students with disabilities who require reasonable academic accommodations should schedule an appointment to discuss specifics with me. It is the policy of Villanova to make reasonable academic accommodations for qualified individuals with disabilities. You must present verification and register with the Learning Support Office by contacting 610-519-5176 or at learning.support.services@villanova.edu or for physical access or temporary disabling conditions, please contact the Office of Disability Services at 610-519-4095. Registration is required to receive accommodations.
· Academic integrity statement with link to Academic Integrity Gateway
last revised: 24-august-2016