MATH 494/598

Mathematical Introduction to Medical Imaging

Fall Semester 2008

Lecture Notes
The original drafts of most of the lecture notes and assignments were proposed by the undergraduate TA Steven Troxler.

Week 1 : Review of Linear Algebra, Analysis, part 1
Week 2 : Review of (finite-dimensional) Linear Algebra, part 2
Week 3 : Review of Linear Algebra, part 3. The Discrete Fourier Transform.
Week 4 : Review of Analysis, convergence and completeness, Lebegue integration
Week 5 : Review of Analysis, L^p spaces, convolution, weak derivatives, intro to distributions
Week 6 : The Fourier Transform , Intro to MRI (guest lecture by Jim Pipe)
Week 7 : MRI physics
Week 8 : Gradient fields and Signal Equation , Sampling (coming up), Non-uniform sampling (seminar talk by Ken Johnson)
Week 9 : Phase and Frequency Encoding (coming up), Excitation and Slice Selection
Week 10 : Off-resounance sources, Gradient Echo and Spin Echo, Image Constrast
Week 11 : The Radon Transform , Modern Aspects of MRI (guest lecture by Jim Pipe)
Week 12: Analysis of 2d Radon Inversion formula, back projection
Week 13: Filter in Radon inversion formula, Hilbert transform, Physics of x-ray CT, CT geometries
Week 14: Radon Transform in MRI
Week 15: Applications of Probability in Medical Imaging
List of Topics
- Review of Linear Algebra
- Metrics, Norms, Inner Products
- Discrete Fourier Transfrom
- Convergence and Completeness
- Basics of Lp theory
- Fourier series
- Convolution and Intro to Distributions
- Fourier Transform (extention to L^2 and tempered distributions)
- Basics of Sampling Theory
- Theory of the Radon Transform
- Basics of CT
- Reconstruction in CT: filtered back-projection
- Basics of MRI: Physics of MR, Magnetization, Relaxation, RF and Gradient Fields, Excitations, Slice Selection, Phase and Frequency Encoding, Bloch equation, 3D Imaging, 2D Projection Reconstruction, 2D FT reconstruction
- More topics in MRI: Reconstruction in MRI: rectilinear sampling and gridding, Contrast Adjustment (T1 and T2 weighting), Fast Imaging Methods, Modern Techniques
- Basics of PET: physics and imaging
- Applications of Probability