In the last 50 years have witnessed an ongoing revolution in image
reconstruction
techniques in fields from astrophysics to electron microscopy and most
notably
in medical imaging. In each of these fields one would like to have a
precise
picture of a 2 or 3 dimensional object that cannot be obtained
directly. The
accessible data is typically some collection of filtered averages. The
problem
of image reconstruction is to build an object out of the averaged data
and then
estimate how close the reconstruction is to the actual object.
In this
course
we introduce the mathematical techniques used to model measurements and
reconstruct images. In this context we cover some of the basic
principles
of
mathematical analysis, the Fourier transform, interpolation and
approximation
of functions, sampling theory, digital filtering, noise analysis, and
image analysis.
The
prerequisites are calculus through math 241, basic mathematical analysis, linear algebra as
well as
basic physics.
Coursework: Problem sets,
and/or computational exercises
will be assigned every week, or every other week; in addition each student will be asked
to prepare a presentation for the class on a subject related to
imaging.
The textbook is Introduction
to the Mathematics of Medical
Imaging, 2nd Edition
by C.L. Epstein. Students should purchase the textbook directly from
Amazon, or SIAM (become a student member of SIAM for free, and get the
discounted member price).
Additional material may be
taken from
the following sources:
- *Principles of Computerized Tomographic Imaging by
Avinash C. Kak
and Malcolm Slaney This book is now available online at Kak and Slaney
- *Computerized tomography: The new medical X-ray technology
by L.A. Shepp
and J.B. Kruskal in Amer. Math. Monthly 1978, pages 421-439
- Image reconstruction from Projections by Gabor T.
Herman
- Foundations of Medical Imaging by Cho, Jones, Singh
- Intermediate Physics for Medicine and Biology by
Russell K. Hobbie
- Radiological Imaging by Barrett and Swindell
- The Mathematics of Computerized Tomography by Frank
Natterer
- *Numerical Linear Algebra by Trefethen and Bau
- Foundations of Image Science
by Barrett and Meyers. This is an encyclopedic treatment of
mathematical methods used in imaging.
- Inside
Out: Inverse Problems and Applications. A collection of
mathematical articles on topics connected to imaging, Gunther Uhlmann,
editor.
- Webpage with software and references on the Non-uniform Fast
Fourier Transform (NUFFT).
