Erkki Somersalo: Introduction to Inverse Problems

Martin Bech: Tomography and image analysis at Max-IV

Ville Kolehmainen: Introduction to Bayesian inversion & MCMC sampling

I will give a brief intro of Bayesian inversion, show a few examples and then explain basics of MCMC using simple class room examples.

Kenichi Kanatani: Introduction to Statistical Optimization for Geometric Estimation

I summarize techniques for geometric estimation from noisy observations for computer vision applications.  I first discuss the interpretation of optimality and point out that geometric estimation is different from the standard statistical estimation.  I also describe the noise modeling and a theoretical accuracy limit called the KCR (Kanatani-Cramer-Rao) lower bound.  Then, I formulate estimation techniques based on minimization of a given cost function: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization as a special case, and Sampson error
minimization.  I describe bundle adjustment and the FNS scheme for numerically solving them and the hyperaccurate correction that improves the accuracy of ML.  Next, Then, I turn to estimation techniques that are not based on minimization of any cost function: iterative reweight, renormalization, and hyper-renormalization. Finally, I show numerical examples to demonstrate that hyper-renormalization has higher accuracy than ML, which has widely been regarded as the most accurate method of all.  I conclude that hyper-renormalization is robust to noise and currently is the best
method. Participants are required to try this numerical experiments themselves for ellipse fitting.

Related reading material (not expected in advance):

Kanatani, K.: Overview of optimization techniques for geometric estimation, Memoirs of the Faculty of Engineering, Okayama University, Vol.47, pp.1--18 (2013). 

Martin Lindahl 1: An introduction to structural biology and single particle cryo electron microscopy (spcm)

Structural biology (in a subcellular perspective) deals with the study of biological form and organization ranging from atomic structures of individual macromolecules to the internal organization of entire cells. Cryo electron microscopy bridges these extremes of length-scales as can provide structural reconstructions of complexes of macromolecules with a spatial resolution that approaches atomic detail. Here we will give an overview of the role of spcm in structural biology and discuss the basic techniques involved.

Martin Lindahl 2: Principles and practices in spcm image processing

The process of going from EM images to visually interpretable 3D volumes typically involves steps such as 2D classification and 3D alignment and reconstruction with tomographic (-like) techniques. We will discuss some of these techniques with a perspective towards inverse problems and try our hand at a basic implementation in Matlab®

Suggested reading (not required):

JF. Frank Three-Dimensional Electron Microscopy of macromolecular assemblies: Chapter 5.1-5.9:  193-259.

Per Christian Hansen: Algebraic Iterative Reconstruction Methods

We present a survey of some iterative reconstruction methods for linear inverse problems that are based on the algebraic formulation of the problem, A x = b, with a focus on the ART and SIRT methods. We survey the basic properties of these methods, discuss how and why they work, and demonstrate how to accelerate and stop the iterations. We also illustrate the use of these methods with hands-on MATLAB exercises, using existing implementations of these methods in the packages AIR Tools and Regularization Tools as well as pre-defined test problems.