Tuesday and Thursday, 12:30–2:00pm, 2013 ECEB

In the first part of the course we will introduce the major imaging modalities. We will start with the 2D Fourier transform as it repeatedly turns out to play a central role in understanding how images come about, and then use it to study optical imaging, tomography, and interferometric as well as diffraction-based techniques. We will then move to a study of the abstract notion of an ill-posed inverse problem, where we'll talk about the issues of uniqueness and stability, introduce regularization techniques and apply them to a selection of problems introduced in the first part. In addition to classical techniques we will touch upon sparsity, compressive sensing, and phase retrieval.

2D Fourier transform

Definition and existence

Properties: Parseval identity, Plancherel identity, inverse transform

Linear operators on images, convolution

Uncertainty principle

Physics of image formation / remote imaging for different modalities

Optical imaging

X-ray tomography

Diffraction crystallography

Interferometric Radio Astronomy

Ill-posed inverse problems

Direct and inverse problems

Well-posed problems and ill-posed problems

Regularization techniques

Tikhonov

Landweber iteration

SVD truncation

(all special cases of) SVD windowing

Sparsity-promoting regularization

Applications in deblurring and tomography

Phase retrieval

Charge flipping

Lifting

First-order methods

Week | Slides and notes | Additional material | Reading | Homework assignment |

1 | Lecture 1, Lecture 2 | Blahut 1.1–1.5, 3.1–3.5 | – | |

2 | Lecture 1 | Blahut 3.4, 3.6, 3.7, 3.9 | – | |

3 | Lecture 1, Lecture 2 | Prof. Do's notes on optical imaging | Blahut 4.1–4.3, 4.5 | Assignment 1, Code skeleton, Source |

4 | Lecture 1, Lecture 2 | Python notebook on FBP, Python notebook on matrix representation and the adjoint | Blahut 10.1–10.2 | |

5 | Lecture 1, Lecture 2 | Blahut 8.1–8.7 | Assignment 2, IPython NB, Source | |

6 | Lecture 1 | Prof. Do's notes on iterative methods, Python notebook on alternating reconstruction | Blahut 13.1–13.2 | |

7 | Project information, Some ideas for papers | Some notes on Prony's method, A great survey of array methods including MUSIC | ||

8 | Lecture 1 and 2 | Python notebook on ill-posed reconstruction | Assignment 3, IPython NB, Source | |

9 | Lecture 1 and 2 | Source, Assignment 4, IPython NB | ||

11 | Lecture 1, Lecture 2 (convexity) | Source, Assignment 5 | ||

All submissions will happen over Compass.

50% homeworks

50% final project (25% journal review + proposal, 50% report and code, 25% presentation)