## Medical Image Science: Mathematical and Conceptual Foundations

### (Medical Physics/Biomedical Engineering 573)

##### Instructors: Diego Hernando, PhD, and Sean Fain, PhD

Offered: Fall Semester

This course covers the mathematical fundamentals required for medical imaging science. Prof. Hernando covers fundamentals of signal analysis with an emphasis on Fourier transforms in one and multiple dimensions. Additional topics include noise in imaging and image reconstruction. This is a hands-on course with a combination of theoretical foundations (on the white board) and computational exercises (using a language such as Python or Matlab) on real and simulated datasets. Mathematical concepts are presented in the context of real-world clinical and research challenges.

Below I share the course materials, including lecture notes, Jupiter notebooks, and mybinder links to directly run these notebooks (thank you for the tip, Kristy Wendt!).

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### Materials for the first half of Med Physics/BME 573

Lecture 0: Introduction / Jupyter notebook (launch mybinder)

Lecture 1: Signals in 1D and N-D / Jupyter notebook (launch mybinder)

Lecture 2: LSI systems in 1D / Jupyter notebook (launch mybinder)

Lecture 3: LSI systems in N-D / Jupyter notebook (launch mybinder)

Lecture 4: The Fourier transform / Jupyter notebook (launch mybinder)

Lecture 5: Properties of the Fourier transform / Jupyter notebook (launch mybinder)

Lecture 6: Fourier transforms in N-D / Jupyter notebook (launch mybinder)

Lecture 7: Properties of the Fourier transform in N-D / Jupyter notebook (launch mybinder)/ Extra notebook (launch mybinder)

Lecture 8: Sampling in 1D / Jupyter notebook (launch mybinder)

Lecture 9: Sampling in N-D / Jupyter notebook (launch mybinder)

Lecture 10: Recap of LSI, Fourier, and sampling / Jupyter notebook (launch mybinder)

Lecture 11: DFT and FFT / Jupyter notebook (launch mybinder)

Lecture 12: DFT in multiple dimensions / Jupyter notebook

(launch mybinder)

Lecture 13: Properties of the DFT / Jupyter notebook (launch mybinder)

Lecture 14: DFT and convolution / Jupyter notebook (launch mybinder)

Lecture 15: DFT and image reconstruction / Jupyter notebook (launch mybinder)

Lecture 16: Limitations of the DFT. STFT. / Jupyter notebook (launch mybinder)

Lecture 17: Intro to wavelets / Jupyter notebook (launch mybinder)

## Imaging in Medicine: Applications

### (Medical Physics/Biomedical Engineering 574)

##### Instructors: Diego Hernando, PhD, and Sean Fain, PhD

Offered: Spring Semester

This course covers topics in medical imaging and image processing, including image reconstruction, registration, and segmentation. These topics provide a deeper understanding of medical imaging systems, and are important for both the characterization of existing systems and for the development of novel imaging techniques. Prof. Hernando covers optimization problems and techniques, which have general applications in imaging and beyond, as well as image reconstruction. This course combines a theoretical framework with computational examples and exercises.

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### Materials for the first half of Med Physics/BME 574

Lecture 1: Intro to the course

Lecture 2: Intro to optimization / Jupyter notebook (launch mybinder)

Lecture 3: Review of matrices, vectors, norms, and linear least-squares

Lecture 4: Constrained optimization / Jupyter notebook (launch mybinder)

Lecture 5: Convex optimization (I)

Lecture 6: Convex optimization (II)

Lecture 7: Optimality conditions

Lecture 8: Line search algorithms / Jupyter notebook (launch mybinder)

Lecture 9: Gradient-based algorithms

Lecture 10: Newton algorithms / Jupyter notebook 1 – Newton (launch mybinder) / Jupyter notebook 2 – NLLS and Gauss-Newton (launch mybinder)

Lecture 11: Intro to stochastic algorithms

Lecture 12: Intro to image reconstruction

Lecture 13: Direct image reconstruction methods / Jupyter notebook (launch mybinder)

Lecture 14: Matrix-vector operations as image-based operations / Jupyter notebook (launch mybinder)

Lecture 15: L2-regularized image reconstruction / Jupyter notebook (launch mybinder)

Lecture 16: L1-regularized image reconstruction and compressed sensing / Jupyter notebook (launch mybinder)