### Location

the tutorial took place on Monday 5 October, in the Holiday Inn, Room *Forum 16*

### Announcement

We are pleased to announce a tutorial on Gaussian process methods and their application to medical data, at the 2015 MICCAI conference.

Gaussian processes are a fundamental concept of probability, with a long history (including repeated re-discoveries) and a very broad range of applications. They are particularly useful as a framework for inference on the value of real-valued functions over generic input domains. In this form, they are employed in areas including physics, control, quantitative biology, climatology, robotics, and indeed imaging. Recent developments have also brought to light that, implicitly, they have also been a fundamental tool of numerical mathematics for many decades.

The tutorial will feature a basic introduction to the concept, it's strength and limitations, by Philipp Hennig. This will be followed by an invited talk by Marcel Luethi on applications of Gaussian process regression in medical imaging, and a tutorial by Søren Hauberg, on advanced uses of Gaussian process modelling for inference from medical image data.

## Schedule

08:30 - 09:45 | Introduction to GP regression [Philipp Hennig]. The slides for this tutorial can be found [here]. |

09:45 - 10:30 | Research perspective — GPs for surface and image registration [Marcel Luethi]. The slides can be found [here]. |

10:30 - 11:00 | coffee break |

11:00 - 12:10 | GP for Numerics with application in DTI [Søren Hauberg]. Slides can be found [here] (best viewed in firefox). |

12:10 - 12:30 | Gaussian Processes in Numerics [Philipp Hennig]. Slides can be found [here]. |

## Titles & Abstracts

### Introduction to Gaussian process regression [Philipp Hennig]

*This first part of the tutorial will be a gentle, basic introduction to Gaussian modelling and inference on functions. Only elementary concepts of linear algebra and probabilistic inference are required. The session will begin by pointing out some key properties of Gaussian distributions, then introduce general linear Gaussian (least-squares) regression. This concept will be extended to the nonparametric ("infinite dimensional") case. Some algebraic properties of positive definite kernels will be listed. A second part, after the coffee break, will add a look at some more advanced theoretical aspects concerning the effectiveness and efficiency of Gaussian process (kernel ridge) regression on generic learning problems. The presentation includes generic code examples, which will be made available on this webpage.*

### Gaussian Processes for surface and image registration

*Non-rigid registration is one of the core problems in medical image analysis. Many different methods have been proposed during the last decades. A main differentiation factor of different registration approaches is how they incorporate prior assumptions about the possible deformation between a reference and target object. In this tutorial we show how Gaussian processes can be used to build expressive prior models for registration, which subsume many of the assumptions formulated in the literature.
We start by briefly introducing two core concepts: 1) Gaussian Random Fields, which generalize Gaussian processes to models over vector-valued functions and 2) the Karhunen-Loeve theorem, which gives us a representation of the Gaussian process as a linear combination of orthogonal functions. Using these two concepts, we derive a simple registration algorithm that uses Gaussian processes to model our prior assumptions.
In the main part of the talk we discuss how we can use simple building blocks to construct sophisticated prior models using Gaussian processes. We show how these models lead to interesting registration methods, including non-stationary, multi-scale registration or hybrid landmark/intensity registration. We also discuss the connection to Statistical Shape Models and how this registration approach unifies shape model fitting and non-rigid registration.
We will illustrate all the concepts on a concrete registration example, using the open source software framework scalismo.*

### Using GP regression for estimating tracts in noisy DTI [Søren Hauberg]

*In this session, we will develop a simple numerical algorithm for solving ordinary differential equations (ODEs) using GP regression.
Many estimation and prediction tasks rely on the solution to ODEs that are defined
by noisy data. Unfortunately, standard ODE solvers are fundamentally unable to model uncertainty
in the ODE itself, and we need to develop new machinery. An ODE solver based on GP regression provides a remedy to this issue.
As a guiding example, we will consider the estimation of tracts in noisy diffusion tensor images (DTI).
Here we need to solve ODEs with coefficients dictated by noisy DTI data, which means that we need to solve a set of uncertain ODEs. This will allow us to build a first probabilistic solution to shortest path tractograhy.
At a more fundamental level, the lecture presents the view that even numerical algorithms should be described probabilistically. The basic ideas will, thus, also be applicable to problems in integration, optimization, etc.
At the end of this lecture, you should be able to tell people that differential equations can be easy.
Joint work with: Philipp Hennig, Michael Schober, Niklas Kasenburg, Matthew Liptrot, and Aasa Feragen.*