## Med Physics / BME 573 Medical Image Science: Mathematical and Conceptual Foundations

##### Instructor: Diego Hernando, PhD

Offered: Fall Semester

##### New name starting in 2023: **Mathematical Methods in Medical Physics**

This course covers mathematical fundamentals required for medical physics. The first half of the course covers fundamentals of signal analysis with an emphasis on Fourier transforms in one and multiple dimensions. The second half introduces mathematical optimization, with applications in medical imaging and therapy. 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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### Part I: Multidimensional Signal Analysis

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)

### Part II: Introduction to Optimization in Imaging and Therapy

Lecture 20: Intro to optimization / Jupyter notebook (launch mybinder)

Lecture 21: Review of matrices, vectors, norms, and linear least-squares / Jupyter notebook (launch mybinder)

Lecture 22: Constrained optimization / Jupyter notebook (launch mybinder)

Lecture 23: Convex optimization (I) / Jupyter notebook (launch mybinder)

Lecture 24: Convex optimization (II) / Jupyter notebook (launch mybinder)

Lecture 25: Optimality conditions

Lecture 26: Line search algorithms / Jupyter notebook (launch mybinder)

Lecture 27: Gradient-based algorithms

Lecture 28: Newton algorithms / Jupyter notebook: Newton (launch mybinder) / Jupyter notebook: NLLS, Gauss-Newton (launch mybinder)

Lecture 29: Intro to stochastic algorithms

Lecture 30: Intro to image reconstruction

Lecture 31: Direct image reconstruction methods / Jupyter notebook (launch mybinder)

Lecture 32: Matrix-vector operations as image-based operations / Jupyter notebook (launch mybinder)

Lecture 33: L2-regularized image reconstruction / Jupyter notebook (launch mybinder)

Lecture 34: L1-regularized image reconstruction and compressed sensing / Jupyter notebook (launch mybinder)

Lecture 35: Intro to optimization in therapy planning

Lecture 36: Optimization for IMRT

Lecture 37: Recap of optimization in imaging and therapy