TP3: Statistical Inference and Data-Driven Selection of Regularisation Parameters for Medical Imaging

Subproject leaders

PD Dr Nicolai Bissantz, Prof. Dr Holger Dette

Project researcher

Dr Justin Chown

Contact

tp3@med4d.de

Objectives

Reliable methods for deciding whether an observed feature in an image is real or a random artefact are essential for medical diagnosis.

This subproject develops new methods to distinguish genuine object features from statistical fluctuations in medical imaging. Reconstructing data from medical and biological imaging is particularly challenging because the object itself is observed only indirectly—for example, through a Radon transform in tomographic methods such as PET.

Reconstruction generally requires regularisation parameters. Too little regularisation creates image artefacts that may resemble lesions but are only noise. Too much regularisation, by contrast, removes real irregularities from the reconstructed image.

We will develop methods for semi-automatic selection of regularisation parameters together with statistical inference based on these reconstructions. These methods will support more accurate decisions about whether an irregularity is statistically significant—and therefore likely to be real.

For multidimensional data, such as data with spatial and temporal coordinates, the method will allow separate regularisation parameters to be selected for each dimension. The methods will be validated using real data, in particular in cooperation with TP4.

Inverse statistical regression models are generally written as

$Y_i=(Kf)(x_i)+\epsilon_i,\quad i=-n,\ldots,n.$

Here $x_i$ are the design points, $K$ is an operator, $f$ is a signal that cannot be observed directly, and $\epsilon_i$ are error terms. Estimators in this model require a regularisation parameter because the operator $K$ generally has an unbounded inverse. The regularisation parameter determines the degree of smoothing and quantifies the balance between confidence in the observed signal (and thus the signal-to-noise ratio) and prior knowledge about the smoothness of the true signal.

Our project will develop data-driven methods for selecting regularisation parameters in typical image reconstruction and visualisation problems in medicine and biology.

Input (from partners and subprojects)

Models for time-continuous measurements and regularisation parameters (TP1, TP2)

Data (TP4)

Clinical PET problems and evaluation of practical methods (all industry partners)

Definition of relevant decision problems for PET data (Novartis)

Output (to project partners)

Methods for selecting regularisation parameters and for statistical inference (all subprojects)

Application of data-driven parameter selection methods (TP4)

Algorithms for clinical studies (industry partners)

Previous work

The Bochum group has worked for many years on statistical inference for inverse problems. Our publications include fundamental results on estimators, selection of their regularisation parameters, and construction of uniform confidence bands for statistical inverse problems.

Another focus of our work is the study of empirical residual distributions: the differences between observations and predictions from a nonparametric regression model. These distributions can be used to validate model assumptions statistically. For example, we use empirical processes based on residuals to test symmetry of the error distribution and to test assumptions such as normality or homoscedasticity.