This paper addresses the problem of anomaly detection in hyperspectral images. We propose and exploit a data model to establish the link between two main approaches in the area of anomaly detection, which are Hypothesis Testing (HT) and Projection Pursuit (PP). We show that joining these two approaches enables to overcome some limitations of each method taken separately. Indeed, the resulting detection algorithm, namely ACP (for Anomalous Component Pursuit) has an asymptotically constant false alarm rate, like HT-based detectors, and enables anomaly spectral discrimination, including the estimation of the number of classes. We assess the ACP algorithm on real-world data, in terms of detection and discrimination, and discuss some theoretic limitations.