What is Statistical Learning

Biomedical imaging represents a practical and conceptual revolution in the applied sciences of the last thirty years. Two basic ingredients permitted such a breakthrough: the technological development of hardware for the collection of detailed information on the organ under investigation in a less and less invasive fashion; the formulation and application of sophisticated mathematical tools for signal processing within a methodological setting of truly interdisciplinary flavor. A typical acquisition procedure in biomedical imaging requires the probing of the biological tissue by means of some emitted, reflected or transmitted radiation. Then a mathematical model describing the image formation process is introduced and computational methods for the numerical solution of the model equations are formulated. Finally, methods based on or inspired by Artificial Intelligence (AI) frameworks like machine learning are applied to the reconstructed images in order to extract clinically helpful information. Important issues in this research activity are the intrinsic numerical instability of the reconstruction problem, the convergence properties and the computational complexity of the image processing algorithms. Such issues will be discussed in the following with the help of several examples of notable significance in the biomedical practice.