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Credit:
1 course unit
Elective course
Catalog Description:
In the last 25 years there has been a 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, which cannot be obtained directly. The data that is accessible is
typically some collection of weighted 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. As a simple representative
case we study transmission X-ray tomography (CT).In this context we cover
the basic principles of mathematical analysis, the Fourier transform,
interpolation and approximation of functions, sampling theory, digital
filtering and noise analysis.
Prerequisites:
Mathematics through multivariate calculus (Math 241), as well as
some familiarity with linear algebra and basic physics.
Textbook(s) and/or
other Required Material:
Introduction to the Mathematics of Medical Imaging, by Charles L.
Epstein, Prentice Hall, 2003.
Topics Covered:
- Mathematical models and measurement.
- Linear equations and an introduction to
function spaces.
- Integral transforms: the Fourier
transform, the Radon transform, the Abel transform, convolution.
- Sampling and digitization, interpolation
and approximation.
- Nyquist's theorem and basic concepts of signal
processing.
- Implementing shift invariant filters.
- Imaging hardware and the basic algorithms
of image reconstruction.
- Analysis of systematic artifacts.
- Introduction to probability and statistics
- Analysis of noise in tomography.
Class/Laboratory Schedule:
Lecture: 3 hr/week
Course Objectives:
The goal of this course is to introduce the mathematical techniques underlying most signal and image
processing in a manner that will allow the students to see the big
picture. While the course is constructed around the analysis of X-Ray
computed tomography, the intention is to provide tools that can be
applied to essentially any imaging modality.
Contribution towards Professional Component:
100% Engineering science
Contribution towards
Program Outcomes:
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Multidisciplinary
Ability
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High
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Problem Solving
Approach
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High
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Problem Solving
Methods
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High
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Experimentation
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Low
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Design
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Low
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Professional
Orientation
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Low
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Person(s) Preparing Description and Date:
Charles L. Epstein
August 2007
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