Description: The main goal
of this course is to expose students to the mathematical
theory of Fourier analysis, and at the same time, to
some of its many applications in the sciences and
engineering. In particular, the Fourier transform arises
naturally in a number of imaging problems as in the
theory of diffraction, magnetic resonance imaging (MRI),
computed tomography (CT) and we shall explain how this
happens. This course covers a broad range of topics and
might be of interest to mathematicians and engineers
alike.
Prerequisite: This is an
upperdivision undergraduate and/or lowerdivision
graduate course. Some prerequisites include linear
algebra (Math 104), real analysis (Math 115) and
probability theory (Stats 217). Assignments would
typically involve a fair amount of scientific
programming in any language you like (e.g. Matlab)
together with more classical exercises.
Syllabus:
Fundamental concepts of Fourier analysis
 Continuoustime Fourier transform, Parseval
identity (Plancherel theorem), inverse Fourier
transform
 Fourier series, sampling of bandlimited
functions, Shannon's sampling theorem, aliasing
 The
Fourier transform and time/space invariant operators,
convolutions
 Fast Fourier transform (FFT) and nonuniform
FFTs
 The WeylHeisenberg uncertainty principle
Selected applications
 Spectral representation of stationary stochastic processes, Wiener filtering
 Fourier optics: theory of diffraction, Xray
crystallography and phase retrieval problems

The mathematics of computed tomography (CT) and
magnetic resonance imaging (MRI)

Short introduction to compressive sensing
Textbooks:
 A First Course in Fourier Analysis by D. Kammler, Cambridge
University Press, revised edition, 2008. (Required)
 Introduction to Fourier Optics by J. W. Goodman, Roberts and
Company Publishers, 2005. (Optional)
 Fourier Analysis by T. W. Korner, Cambridge University Press,
1989. (Optional)
 Principles of Magnetic Resonance Imaging
by Dwight Nishimura, 2010. (Optional)
 Fourier Series and Integrals by H. Dym and H. P. McKean, Academic
Press, 1972. (Optional)
 Introduction to the Mathematics of Medical Imaging, Second
Edition, by C. L. Epstein, Society for Industrial and Applied
Mathematics (SIAM), Philadelphia, PA, 2008. (Optional)
Handouts:
All handouts will be posted online.
Course assistant and office hours:
 Erik Bates () Office hours W 2:304:30, 380381F
Grading (tentative):
 Homework assignments: 50%
 Homework assignments will generally be distributed on
Wednesdays and are due in class the following
Wednesday.
 Late homeworks will NOT be accepted for grading
(medical emergencies excepted with proof).
 There will be about 4 or 5 assignments; the lowest score will be
dropped in the final grade.
 It is encouraged to discuss the problem sets with others, but
everyone needs to turn in a unique personal writeup.
 Final project: 50%. Students will have the freedom to
select a project from a list according to their own
interest. Students can also define their own project after
communicating with the instructor.
